an i1westigation of small smllple pro~ties … scope and method of sampling ••••• 2.2...
TRANSCRIPT
AN I1WESTIGATION OF SMALL SMllPLE PRO~TIES OFZAHL! S ESrnl1ATE OF A GENERAL MARKOV PR(j~SS MODEL FOR
FOLLOW UP STUDIES
Kathleen Elizabeth Wh*te
This research was supported in part by National Institutesof Health Grant CYP-3042, U.S. Public Health Service GrantCA-06122-03 and Contract SA-43-ph-1932, and University ofWashington ResGnrch Computer Laboratory Grant USR 30.
Institute of StatisticsMimeograph Series No. 391May, 1964
iv
TABLE OF CONTENTS
Page
• 0 • • 0 • e _ •LIST OF TEXT TABLES • •
LIST OF APPENDIX TABLES
LIST OF FIGURES • • • •
. . .
. . . . . . . . . .• '" 0 0
• • • •
· .1. ZAm..'S ESTIMATES OF A GENERAL MARKOV PROCESS MODEL FOR
FOLLOW-UP STUDIES • • • • • • • • • • • • • • • . . 1
1.1 Introduction • • .'... • • • • •1.2 Mathematical Definition of the General Markov
Process Model • • • • • • • • • • • • • • •1.3 Description.of a Particular Model for Follow=up
Studies to to 0. ... 0. <0 0 0 to 0. ~ 0 • 0 0 0 0- 0. " 0 . . .
1
2
4
66788
1011
. .. . .
. . .. . .
1.4 Maximum Likelihood Estimates of Zahl for the GeneralMarkov Process Model ••••1.4.1 Sample Design •••••••••••••••1.4.2 Estimation Procedure • • • • •
1.5 Asymptotic Properties of the El;!timates •••1.5.1 General Model. • • • • • •1.5.2 Particular Model ••• • •••••••••
1.6 Merit of the Estimates in Follow-up Studies • • • •
2. MONTE CARLO INVESTIGATION OF SMALL SAMPLE PROPERTIES OFZAHL ' S ESTIMATES • • • • • • • • • • • • • 12
2.1 Scope and Method of Sampling •••••2.2 Deficiencies of the Estimates for Small Samples ••
2.2.1 Incomplete Samples • • • • •••2.2.2 Divergent Samples • • • • • • •2.2.3 Finite Termination of the t Series ••••2.2.4 Negative Ai1 Values • • • • ••••••••••2.2.5 Discont±nui~y of Pij(t') and the Effective
Sample Size ••• • • • • • • • • • • •2.3 Definition of Con~itional Probability Distr~butions
of (Pij(t')}, (Aij ) " • • • • • • • • • • • • • •
3. EVALUATION OF THE PROBABILITY GENERATING FUNCTION • • '. • • •
121414182122
24
25
28
3.1 Discussion • • • • • • • • • • • • • • • • • • • • •• 28·3.2 Definition of the Probability Generating Function. 283.3 Distributions Obtained from Probability G~nerating
. Function Evaluation • • • • • • • • .". • • • • • 29
v
TABLE OF CONTENTS (Con~inued)
Page
3.4 Sampling ?istributions of the Estimates (Pij(l.O»),for r = 3, N S 5 • • • • • • • 0 0 • • 0 • • • • • •• 303.4.1 Probabilities of Types of Samples' 0 •• o. 303.4.2 Distributions of Effective Sample Sizes • • •• 303.4.3 Marginal Unconditional Distributions of
(Pij (1.0») • 0 ••••• 0 • • • 0 37
3.4.4 Conditional Bias by Sample Type • • • • • • •• 423.4.5 Conditional Covariance Matrices by Sample Type. 463.4.6 Conditional Mean Product Error Matrices 52
4. RESULTS OF MONTE CABLO INVESTIGATION 0 . . . . . . . . . .0. 53
. . .
4.3.4 Bias .... ~ ., 0' ••• 0 • 0 • '"
4.3.5 Variance 0 .. 0- • "" 0 • 0 • If ••••• 0 ••••
4.3.6 Covariance • • • • • • '. • • • • • • • •4.3.7' Mean. Square Errors • • • • 0 • • • • • • • • • •
4.1 Choice of Methods of Analysis of Results • • •4.2 Multivatiate Analysis ••••••• 0 0 0
4.2.1 Mean Vector and Covariance Matrix o.
4.2.1.1 Description of Test • 0
4.2.1.2' Results •••••• 0 •
4.2.2 Independence of (Plj ) and (P2k) ••
4.3 Univariate Analysis •••••••••••••4.3.1 Distributions of Effective Sample Sizes4.3.2 Range and Frequeapy of Negative (tij )
4.3.3 Kolmogorov-Smirnov Test of MarginalDistributions of (Pij ) and (tij ) •
· .. . . ..• • 0
. .'
535555555659
616165
67
77858990
95
9595959697
9Q100101104104105105lQ7107
· .· .
• • 0-
• • •• • • 0' • • • • • ..• 0
5.1 Discqssion ••••••••• 0 •••••
5.1.1 Generality of Results of the Study •••••5.1.1.1 Model •••• • ••5.1.1.2 Initial Condition ••••••••.5.1.1.3 A Parameter Matrix • • • •5.1.1.4 Number of Observations per Patient,
r, and Length of Observation TimePeriod, to ••••••••••
5.1.1.5 Sample Size •• 0 0 • • • • •
5.1.2 Considerations in Sample Design ••••••5.1.3 Statistical Inference from Zahl's Estimates ••
5.1.3.1 Sample Size,Less,Than Twenty •••••5.1.3.2 Sample Size Twenty to Fifty •••••5.1.3.3 Sample Size Fi,fty t~ Two Hundred •••5.1.3.4. Sample Size Two Hundre,d •• • •
5.1.4 Indications for ,Further Research ••••••••.
DISCUSSION AND CONCLUSIONS •5.
vi
TABLE OF CONTENTS (Continued)
Page
· . . . . . . . . .· . . . . . . . . .
. . . . . . . .
5.2 Conclusions •••••••••• • •5.2.1 Sample Size Less Than Twenty5.2.2 Sample Size Twenty •••5.2.3 ~ample Size Fifty •••••5.2.4 Sample Size Two Hundred ••
108108108109109
111. . . ..
Q • • • •
. . .
. .
· . .• 0' • • 0 •. .. . .• •LIST OF REFERENCES6.
7• APPENDICES. • • • ., • • • • • • • • • 41 • • • • • • • • • 113
7.1 Asymptotic Covariance Matrix Theorem. • • • • • • • •• 113
7; 2 Properties of [p*( t i)] S and Bounds on the RemainderAfter a Finite Number of Terms of the A Series • • •• 115
7.3 Evaluation of the Probability Generating Function .••• 1277.3.1 Method of Evaluat ion • • • • • • • • • • • • •• 1277.3.2 Accuracy of the Evaluation and Magnitude of
the Computations • • • • • • • • ~ • • • • •• 1297.4 Monte Carlo Parameter Tables • • • • • • • • • • • • •• 1327.5 Probability Generating.Function Evaluation Tables .• 1417.6 Monte Carlo Data Tables ••••••••••••• 1507.7 Multivariate Tests • • • • • • • • • .. .. • • • • • • -.. 196
7.7.1 The Hypothesis that a Mean Vector and aCovariance Matrix Are Equal to a GivenVector and Matrix • .. • • • • • • • • • • • • 196
7.7.2 The Hyp~thesis that Two Sets of ThreeVariates Each Are'Independent.. • • 196
•
vii
LIST OF TEXT TABLES
Page
2.1. Frequency of Monte Carlo samples by t;ype and by t',r, N '" .. 9 0 0- 0 <I • 0 0- (I. '" 0- 0- 0 0 <I 0- • 0- Q 0- 17
3.1. Joint probability distribution of effective sample sizesIll.' n2 ., from 'probab:l,lity generating function
eval~ation for t' equal 1.0, r equal 3, N equal 2 33
3.2. Joint probability distribution of effective sample sizesnl ., n2., from probability gene~ating function
evaluation for t' equal 1.0, requa1 3, N equalS 34
3.3. Joint relative frequency distribution of effective samplesizes xl1.' n2., from 100 Monte Carlo i samples ~or t' '
equal 1.O,r equal 3, N equalS. • .'. • . • • . 35~I - I
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.
Values of the likelihood ratio test statistic of thehypothesis of asymptotic mean vector and covariancematrix of convergent (P.j(t U
)) by r, t ' , and: N •• '.,; 1,
Values of the likelihood ratio test statistic of thehypothesis of asymptotic mean vector and covariancematrix of unmodified '(~ij) by r, t I ,'~nd N • • • • •
Values of the likelihood ratio test statistic of thehypothesis ofjasymptotic mean vector and covariancematrix of,modified (tij ) by r,:t ' , and N •.....
"
Values of the likelihood ratio test statistic of thehyp0t;hes:Ls of zero covariance of (Plj(t l ),P2k(t,')),by t , ,r1 .and No 0- 0- 0- " !o 0 Q " 0- 0 0- 0 0- '" " 0- 0- •
Valu~s of the Kolmogorov-Smirnov test statistic of thehypothes,isof ,normal margina]} distributions of theeffective sa~ples sizes with mean~ and variancesdetermined from 'probability generating functionevaluat:i:on for't' equal 1.0, by sample size and by r ••
Range and'~frequency of negative values of (ri .) byI J'
t '; r, ~t;ld N q (I., 0- (I. o· '" 0. 00 0- 0 '" • 0 0- 0- _0 G
57
58
60
62
64
66
4.7. Values ofdthe Kolmogorov-Smirnov test statistic ofhypothesis of normal marginal distributions o~
convergent (Pij(l.O)) with asymptcltic means and
variances, by sa~ple size,and by r •• ; •••.
the
o • • • 72
viii
LIST OF TEXT TABLES (Continued)
Page
4.8. Values of the Ko1mogorov-Smirnov test statistic of thehypothesis of normal marginal distributions of themodified (tij ) ~ith asymptotic means and variances,by sample sJ.ze and by r, for t' equal 1.0 ••• · . . 73
76· . .
4.9. Values of the Ko1mogorov-Smil'llov test·statistic of thehypothesis of normal marginal distributions of theunmodified {tij ) with asymp~oti7 means and variances, bys~mp1e size and by r, for t equal 1.0 • • • • • • •• 74
Values of the Kolmogorov-Smirnov test statistio of the-. hypothes:isof normal marginal distribut,ions of
convergent (Pi 0 (t')) with asymptotic m~ans andvariances., 1>:1 J sample size and by t', for r equal 7.' f I
4.10.
4;11. Values of the Kolmogorov-Smirnov test statistic of thehypothesis of normal marginal distributions of
c modified (ti o ) with asymptotic means and variances,
by sample J size.and by t', for r equal 7 •••., I · ... 78
4.12. Values of the Kolmogorov-Smirnov test statistic of the.nypothesis of normal marginal distributions of
.. uT11llodified > {tij } wit1:La~ymptot1c means and variances,by sample size and by t , for r equal 7 ••••••• 79
4.13.
4.14.
4.15.
4.16.
4.17.
4.18.
Estimates of coefficients of bias regres~ion on reciprocalof sample size, and significance of differepce from zero,for convergent (Pij(t P
)}, by t P and r ••••••••
Estimates of coefficients of bias reg~ession on recipro~al
of sample size, and significance of difference from zero,for modified ftijh by e and r •• • • • • • • • • •
-2Estimates of coefficients of variance regression On N ,.and significance of difference from zero, for convergent(Pij (t ') J, -by t.' and r • • • • • • • • • • • • • • • • • •. ' .' '. . -2
Estimates of coefficients of variance regression on N ,and significance of difference from zero, for modified(
0«. }. , ' .Ai co J by t and _r 0' 0 -0' 0- 0- 0. 0 0- 0 0 0 0- 0- 0" it 0 • • • •J . .. '.. -2
Estimates of coefficients of covariance regression on N ,and significance qf difference from zero, for convergent(pij ( t i ) , Pik(t P )), by t i and r • • • • • • .'. • • • • • •
-2Est~ates of coefficients of covariance regression on N 1
an4 significance ·of difference from zero, for modified(tij'~ik)' byt' and r ••••••••••••••••••
83
87
88
91
92
LIST OF TEXT TABLES (Continued)1
ix
Page
4.19.
4.20.
. -2Estimates of coefficients of covariance regression on N ,and sign~ficance' of differencef~om zero, for modified .(t
lj,t2k), by t' and r
l•• ' .••• ' •• ~ •••••• '•••
Estimates of mean square errors of modified and unmodified..(t
ij) from convergent' samples in'sets of 50 for r equal
7, 'l:>y t' and by sample size •••••••••••••••
93
94
5.1. Expected effective sample sizes, probability of anincomplete sample, .f1nd transit,ion probabilities forselected. values of r and,t I. • •• • • • • • • • • • . . . 103
•x
LIST OF APPENDIX TABLES
Page
7.3.1. Magnitude of output and total probabilities ofprobability generating function evaluations fort equal 1.0 e • • e e • • • • • • • • • • . . . 130
7.4.1. MOnte Carlo generating matrices • • • • 0 • .0 • • • • 133
7.4.2. Asymptotic covariance of (Pij(l.O)}, and of ~odified
(~ijJ for r equal 3 • • • e • • • • • • • • • • • • •• 134
7.4.3. Asymptotic covariance of (Pij(l.O»), and of modified
(tij ) for requal 5 • • • • • • • • • • • • • • • • 135
7.4.4.
7.4.5.
7.4.6.
7.4.7.
Asymptotic covariance of (Pij(l.O», and of modified
(tij ) for r equal 7 • • • • • • • • • • • • • • • • •• 136
Asymptotic covariance of (Poj(0.2», and of modified(ti 0) for· r equal 7 • • .1.. • • • • • • • • '. • • • 137J .
Asymptotic covariance of (Pij(2.0», and of modified
(tij ) for r equal 7 • • • • • • • ~ • • •• • • •• 138
ProbabJ.lityof an incomplete sample [h(r,e)]N •••••• 139
7.4.8. Expected effective sample sizes by tV and r •• . . . . . 140
7.5.1. Means, variances, and covariances of effective samplesizes nl ., n2., determined from ~robability
generating'function evaluation for t' equal 1.0, by r. 142
144
7.5.2. Bias of, (Pii(l.O») determined from evaluation of theprobabi11ty generating function for r equal 3 .' • •• 143
7.5.3. Convergent sample conditional covariance of (Po j (1.0»for r equal 3 by sample size ••• 0 •••• : •••
. .
7.5.4. Complete sample conditional covari,ance of (Pij (1.0»for r equal 3 by sample size •• • • • • • • • • 145
. .7.5.5. All samples covariance of (PIo{l.O» for r equal 3by sample size ••••••J ••••••• e •• 146
7.5.60• Convergent sample conditional mean product errors of
(Pij(l.O») fOr r equal 3 by sample size •••••• 147
xi
LIST OF APPENDIX TABLES (Continued)
Page
7.5'.1. Complete sample conditiollal mean product errors of(Pij(l.O») for r equal 3 by sample size. • • • 148
7.5.8. All samples mean product errors of (Plj(l.O») for requal 3by sample size • • '. • • • • • • • 149
7.6.1. Means, variances, and covariance of effective samplesizes for t' equal 1.0, r equal 3, by sampl~ size.. 151
7.6.2. Means, variances, and covariance of effective samplesizes fort' equal 1.0, r equalS, by sample size 152
7.6.3. Mean~variances, and covariance of effective samplesizes for e' equal 1.0, r equal 7, by sample size . . 153
7.6.4. Estimates of bias of (Pij(l.O») and modified (~ij)
from convergent samples in sets of 50 for r equal 3,.by sample size ••••••• • • • • • • • • • • 154
7.6.5. Estimates of bias of (Pij(l.O») and modified (~ij)
from convergent samples in sets of 50 for r equalS,by sample size •• . • . • • • • • • • • • • • • 155
7.6.6. Estimates of bias of (Pij(l.O») and modifie~ (~ij)
from,convergent samples in sets of 50 forr equal7, by sample size • • • • • • ~ • 0 • • • • '.' .~ 156
7.6~7. Estimates'of bias of (Pij(~.2») and modifie'd £tij )
from convergent samples in set~ of 50 for r equal7, by sample size •••.••• ' .•••••• !. • 157
7.6.8. Estimates of bias of (~ij(2.0») and modified (~ij)
from convergent samples in sets of 50 for ~ eq~al7, by sample size •••.•.•••••••.• ~ 158
7.6.9•. Estimates of variance of (Plj(1.9)} from convergentsamples in sets of 50 for requal 3, 5, 7, bysample size • • • • • • • 0 • • • • • • 0 • • • • 159
7.6.10. Estimates of variance of (P2j(1.0»). from convergent
samples in sets of 50 for r equal 3, 5, 7, by sam~le
size • • • • • 0 _,' • '. •• 160
xii
LIST OF APPENDIX TABLES (Continued)
Page
from
. .
7.6.11.
7.6.12.
7.6.13.
7.6.14.
Estimates of covariance of (PlJ(LO),Plk(LO)} from
convergent samples in sets of 50 for r equal 3,5, 7, by sample' size • • • • • • • • • • • •
Estimates of covariance'of (P2j(LO),P2k(1.0»): frollf
convergent samples in sets of 50 for r equal 3,5, 7, by sample s~ze • • • • • •• • • • • •
Estimates of covariance of (P12(l.0),P2j(1.0»)
co~vergent samples in sets of 50 for r equal 3,5, 7, by sample size •• ~ •••.•••••
Estimates of covariance of (Pi3(1.0),P2j(1.O») from
convergent samples in sets of 50 for r equal 3,5, 7,-by sample size ••••••••••••
161
162
163
164
7.6.15. Estimates of covariance of (P14 (l.0),P2j (LO)) from
convergent samples in sets of 50 forr equal 3,5, 7, by.sample size • • • • • • • • • • • • 165
7.6.16. Estimates of variance of modified {~ljJ from convergent, samples in s'etsof50 for t' equal LO, r equal
I Co •
3, 5, 7, by sample S1ze •• • . . • . • • • • . .. 16~1
7.6.17. Estimates of variance of modified (~~.) fromconyergent samples in .set~ of SO . ~ for t'equal 11'.0, r equal 3,.S/ 1, by sample,size •. 167,,
7.6.18. Estimates of covariance of modified £t1J
,tlk} fro~
convergent- samples in sets of 50 for t' equal 1.0,requal 3, 5, 7, by sample size • . • . . . •. 168
7 .6,.19~
7.6.20.
7.6.21.
.Estimates of
convergentr equ~l 3,
Esti.mat~s of
. co.nvergentrequal 3,
Estimates of
convergentr equal 3,
covariance 9f modified {t2.'~2k} fromI . J.
samples in sets of 50 for t' equal 1.0,5, 7, by s4mple size ••.•..••
c~variance of modified (~l2'~2j)- from
samples in sets of SO for t' equal 1.0,S, 7, by sample size••••••.••
covariance of modified (~l3,t2j) from
samples in sets of SO for t' equal 1.0,5, 7, by sample size .•••••.•
169
170
171
xiii
LIST OF APPENDIX TABLES (Continued)
Page
7.6.22. Estimates of covariance of modified (~l4'(~2j) ·from
convergent samples in sets of 50 for t' equal 1.0,r equal 3, 5, 7, by sample size. . 172
t'.6.23. 'Estimates of variance of (Pij(O.2») from convergent
samples ill sets of 50 for r equal 7, by samples 1ze 0 0 0' 0 0 0' 0 0 III I) " 0 0 1;1 0 9" " 0 0 • • " 173
7.6.24. '. Estimates of C9v~riance of (Pij(0.2),Pik(o.2)) from
convergent samples in sets of 50 for r equ~l 7, bysample size • • • . . • • • . • • . . . . . . . . 174
7.6.25. Estimates.c. f c.ovariance O.f (P.lj(O~2),P2k(0.2). }.. ' fromconvergent samples in ~ets of 50 for r equal 7,
-by sample size . . • • . • 175
7.6~26. Estimates of variance of modified (rij ) from
convergent samples4n' sets of SO for t l equal ~.2,
r ~qual 7, by sample size . • . • . •. ~ • • • 176I
7.6.27. Estimates of c~varianceof modified (tij,tikJ from
convergent samplis in sets of 50, for t' equal 0.2,r equal 7, by s~mple size. . • • . . . • . • 177
7.6.28. Estimates of c~var~ance of:modified (tlj ,t2k) from
convergent samples in sets of 50 for t l equal 0.2,r' equal 7, by ~ample size • . . • . 178
7.6.29. Estimates of variance of {Pij(2.0») fro~ couyergent
s~mples in sets of 50 for requal 7, by sample size 179
].. 6 .30. Estimates of covariance of (Pij (2.0),Pik(2.0») from
convergent samples in set~ c~ 50 for! equal 7,.by sample size .' . • • • • • • • • • . • . . t • 180
7.6.31. Estimates of covariance of (Plj(2.0),P2k(2.0)~ from
convergent samples in"sets of 50 for r equal 7, bysample size 0 0' • 0 " 0 0' 0' 0 0 0 .'" 0 0 • .:~.. 181
7.6.34. Estimates of variance ~f mOdified{~ij) from convergent
samples in' sets af 50 for t I equaL 2.,0, r equal 7,.by sample size • ,'.. 182
xiv
LIST OF APPENDIX ~ABLES (Continued)
Page
7.6.33. Estimates of covariance of modified (tij,tik) from
convergent 'samples in sets of 50 for t' equal 2.0,r equal 7, by sample size, • . . . • • • . • . . •• 183
7.6.34. Estimates of covariance ~f mo~ified (tlj ,t2k) from
convergent samples in sets of 50 for t' equal 2.0,, r equal 7, by sample she • • • • . • . . • 184
7.6.35. Estimates of mean square error
convergent samples in sets of7, by sample size • . •
of (Pl j (1.0)} from
50 for r equal 3, 5,. " " -~ " -. . . . . 185
7.6.36.
7.6.37.
Estimates of mean square error of (P2j (1. O)}, from
convergent samples in sets of 50 for r equal 3,5, 7, by sample size • . • • ...•• • .
Estimates, of ,mean product error of (Plj(l.,O),Plk(l.O»)
from convergent samples in sets' of 5Q f\or r equal3, 5, 7, by sample size. • . .. • •....
186
187
7.6.38.
7.6.3~.
Estimates of mean product error of (P2j(l.0),P2k(l.0)}
from convergent samples in sets of 50 for r equal3,5,7, by sample size ••.•.•.•••...
Estimates o~ mean product error of (P12(1.0),P2.(1.0»), J
from convergent samples in sets of 50 for r equal3, 5, 7, by sample size ...•.••.....•••
7.6.40. Estimates of mean proQ.uct' error"of (P13 (l.O)'Pij (l.0))
from convergent samples in sets of S~ for r equal
3, '5, 7, by sample size ••••
188
189
190
7.6.41.
7.6.42.
7.6,43.
Estimates 'of mean product error of (P14 (1.0),P2J(l.O»)
ftomconvergent samples in sets of 50 for r equal3,5, 7, by sample s,he • . • . •.. ••••
Estimates of mean square error of modified (tlj )
from convergent samples in s~ts of 50'for t' equal1.0, r equal 3, 5, 7, by sample size
Estimates of mean square errOr of modified (t2j ) from
conve;rge,nt samples in sets of 50 for t' equal. 1.0,r equal 3, 5~ 7,by s~mp,le size • .•... ". . •
191
192
193
xv
L~ST OF:.PPENDIX TABLES (Continued)
Page
7.6.44. Estimates of mean square error of (Pij (0.2)) from
convergent s,amples in sets of 50 for r equal 7,by sa~ple size • . . • . . • . • . . . 194
7.6.45. Estimates of mean square error of (Pij (2 .On from
convergent samples:ln sets of 50 for r equal 7, bysample size "...... ,. • . .!. . . . 194
7.6.46. Estimates of mea~ s~uare ~rror ~f modified trij )
from converg«!p.t s8ptples in sets of 50 for t' equal, . 0;,2, requal 7', by sample s~ze . .'. .. . • .. 195
7 .. 6.47. .Estimates of mean squ~re ~rror, of .mo~ified (~ij)
, from convergent samples in sets' of 50 ',' for t I equal, 2.0,' 'l;equal 7, by 'sample size. '. ,.' .. ~ . . 195
xvi
LIST OF FIGURES
Page
3.1. Probability of types of samples for t' equal 1.0,. r equal 3, by sample size •. . • . • • • 31
3.2. Marginal probability distributions of effective sample,izes) nl ' n2 ' for t' equal 1.0, r equal 3, N equal2, 5 . .•. . : e • • 0 9' 0- " • (1 0 " 0 " • " • • • • 36
3.3.
3.4.
3.5.
Unconditional, of P
12(1.0)
of sample •
l4lconditionalof P13 (1.0)
of sample •
Unconditional'o,f P23 (1.0)
of samp~e •
<
marginal cumulative probability distributionsfor r equal 3, by sample size and by typ~
marginal cumulative probabilitydist~ibutions; .' "
for r equal 3, by sample' size ~nd by type
marginal cumulative probability, distributionsfor r eq~al 3, by!sample size and by type
39
41
3.6. Percent bias of P12 (1. OJ and PIA(1. 0), SA any absorbing
state, for r equal 3, by reciprocal of sample size, bytype of sample • • • • • • . .. • . • .'. .. . • • .'. " 7 43
3.7 • percent bias of 1>21 (1.0) and P2A(l.0), SA any~bsorbing
state, for r equ8+ 3, by reciprocal of sam~l~ size, )by type of sample • . • .'. • . • • • • • . . • 44
3.~. Varianc~ of P13(1.0) for r, equal,?' by recipro~~l ofsample'size, by type of sample •• • " . . " . \ 47
3.9., Variance of ~23 (1.0) for r equal 3, by reci,procal ofsa,mple'1 size, by type of sample ' • • . • • 48
3.10. Covariance of P12(1.0),P13 (1.0) for r equal 3, by
reciprocal of sample size, by type of sample • . . • •• 49
3 .11~ Covariance of P13 (1.0) ,P14 (1.0) for r equal 3, by
reciprocal of ,sample size, by type of sample • . • .•• 50
3.12. Covariance;of P21(1.0),P23 (1.0) for r ~qual 3, by
reciprocal of sample size, by type of samph • • • • • • I 51
..
LIS~ OF"~GURES (Continued)
4.1. Asymptotic normal_.aad Monte Carlo marginal cumulativedistributions of ~Z3' t' equal 1.0, r e~ual 7,N equal 5J. 10 9 • .'. ~ • ... ill 9' • • • 9' • e ., • .. •
4.2. Asymptotic normal and Monte Carlo marginal cumulativedistributions' of ~23' t' equal 1.0" r equal, 7,N equal 20, 507 . . II • • r> .. .. 0 • • 10 • 0 9' ... .:.
4.3. , Asymp~otic normal and Monte Carlo marginal cumulativedistributions of ~23" t I equal 1.0, r equal 7,,' .N eq~al 100,200 . go ·0 9' .. 0 • • • • • .• .. • • • •• •
xvii
Page
69
70
71
1. ZAHL'S ESTIMATES OF A GENERAL MARKOV PROCESS MODEL
FOR FOLLOW-UP STUDIES
1.1 Introguction
A Markov process model for follow-up studies of cancer or
other disease was introduced by Fix and Neyman (1951). Previously,
survival rates or curves had been used as measures of treatment
efficacy. Actuarial life table methods of analysis were generally
employed; in some cases p,arameters of normal or log-normal survivai
curves were estimated by the method of maximum likelihood (Boag, 1949).
A common problem in follow-up studies is the loss of some
patients before termination of the study,.due either to death from
causes other than the disease under study or inability to trace the
patient. Various methods of weight~ng the incomplete data were
advocated and used. Each method required questionable assumptions and
none w~s entirely satisfactory.
Apart from the technical deficiencies, the actuarial methods of
analysis in ~ollow-up studies utilized only information On the life or
death of a patient~ In the~resence of disease the quality of l~fe may
range over a wide spectrum from complete remission of the disease to
the moribund condition and the end point of death. To the clinician
investigating treatment efficacy, and certainly to the patient himself,
quality of life at any time following treatment is vitally important.
Although the spectrum over which a patient's condition may vary is
2
actually continuous, the clinician's ability to differentiate will
generally be limited to some set of intervals of the range. !Such a set
of defined intervals constitute the states in which a patient may be
observed. The most important advantage of the stochastic model pro-
p,osed by Fix .and Neyman is theprovhion for analysis of data as
refined as the clinician is able to observe with reliability. The. '1··'
par~icular model which they presented in detail was~ however> simpli-
fied to the four states of death from cancer, diagnosed,cancer, normal
life and death from other causes or untraceed. The 'best regular
asymptotic~lly normal' estimates 'of the parameters which they obtained
were applicable only to this special case. Estimates of the ,parameters
of more complex moedels can be obtained by the same methoed, but: each
model considereed requires a separate solution.
Zahl (1955) obtained maximum likelihooed estim,ates of the
parameters defining the general, time statiOnary, continu0l.ls Markov
process with" finite number of states. He examined the asymptotic
properties of the estimates aned proposed tests of several types of
hypotheses based on the asymptotic ,properties. The small sample
properties of Zahl's estim,ates are investigateed in this thesis.
1.2 Mathematical Definition of the
General Markov Process MOedel
Let Si' i ... 1,12, ... ,k, edenote a finite set of states. Define
.Pij(u,U+t) =: Prob (Sj at u+t giv;enSi atu],.-.
and let P(u,u+t).be the kX k matrix (Pi,j(u,u+t1J. p(u,u+t) is
called a Markov transition probability function matrix if
(i)
(ii)
I Pij(u,u+t)·... 1,
j
i ... j
i ,. j
3
(iii) pim(u,u+t+v) ... I Pij(u,u+t)pjm(u+t,u+t+v).
j
The process is said to have stationary transition probabilities if
(iv) Pij(u,u+t) ... Pij(t), for all u~
The preceding conditions may then be written
(i' ) Pij (t) > 0 I Pij(t) :: 1,,j
t' i ... j"(ii') Pi .(0) ...
J 0 i :Jt j,. ,
(iii') Pim(t+v) =: I Pij(t)Pjm(V).
j
Sj is called an absorbing state if
pjj(t) = 1, for every t > 0,
that is, if transition out of S. is not possible, S. is called a. J.. 1
transient state if
Pii(t) < 1, for every t > 0,
and there is some absorbing state Sj for which
p.. (t) > 0, for some finite t.1J
It can be shown (Doob, 1953) that there exists a unique matrix
A ... (Ai.]' such thatJ"
pet) At=e ,
where the exponential function of At is defined
4
Ate
The elements of A are instantaneous transition rates as may be seen
from the relation
A = lim ~P(t) - I]t-+Ot
The elements of A have the following properties
(i)i r/: j
,i = j
(ii)I Aijj
=0, i = 1,2, ••• ,k •
A together with a statement, in absolute or probability terms, of the
initial states determine the probability laws of the process. Hence,
A is the parameter matrix of the model.
The validity of the assumptions of the Markov property (iii)
and of stationarity (iv) requires serious··consideration in applying the
mathematical model to a disease process. The Markov property, that the
conditional probabilities of transitions from a state 8i do not depend
on the history of the individual previous to the time of entering S.,J.
is basic .to the model. The assumption of stationarity implies constant
instantaneous transition rates. In some disease processes these rates
may well be. functions of the time from onset of the disease.
1.3 Description of a Particular Model for Follow-up Studies
The particular model considered in this thesis consists of a set
of four states, of which two are transient and two absorbing. The
states may be interpreted for a disease process as follows:
5
81 : active disease,
82: remission of the disease,
83
: death due to the disease under study,
84 : death due to causes other than the disease or loss from
observation.
The transitions possible in the model are given schematically below
~8l
8 8.2
1 S3
4
Thus 81, 82 are transient states and 83
, 84 are absorbing states. The
instantaneous rate matrix has the form
An A12 A13 114
AA2l A22 1 23 11.24
D
0 0 0 0
0 0 0 0
For finite t the transition probability function matrix has the form
pu(t) P12(t) P13(t) Pl4(t)
pet)P21(t) P22(t) P23(t) P24(t)
::
0 0 1 0
0 0 0 1
These four states constitute a minimal Markov process model for
a follow-up study. It provides a single step improvement over survival
studies by dichotomizing the state of life into active disease and
remission of disease. The state of loss from the study for any reason
not related to the disease is an essentia~ feature of any model to
e·6
provide valid separation of the effect of extraneous factors. In
practice increased refinement of the clas~ification of the patient's
condition would be expected to yield more sensitive differentiation of
efficacy of treatments. However, intuitively, the information, i.e.,
the sample size, required to yield es~imates with given properties
would increase with the complexity of the model. Consequently, the
simplest model was chosen for investigation.
1.4 Maximum Likelihood Estimates of Zahl for the General ~rkov
Process Model
1.4.1 Sample Design
The estimation procedure proposed by Zahl is based on the
following general: sample design. N individuals are observed at an
initial time, t -= 0, and at r subsequent equidistant time points,
t,2t, •• o',rt, .and only the information on the states of the individuals
at these r+l time points is used to estimate the parameters of the
process. The absolute time of observation need not be the same for
different individuals, only the time period between observationso The
sample maybe drawn from a population in which the individuals' initial
states differ or from on~'in which members are all in the same state
initially. In long-term disease studies the latter will usually be
the case with patients entering the study in a state of active disease.
The investigation of the particular model was limited to samples drawn
from individuals 'initially in 81,
Let 1'11(0) denote the number of ,individuals in a sample of size N
who are initially in 8i and nijthe number of transitions from 8i to Sj
observed among the N individuals at the r observation intervals of
The sample in general yields an initial condi~10n vector (ni(O»), and
an observed transition matrix [nij~ which is a function of N the size
of the sample, r the number of observation points, t U the time interval
between observations, and A the parameter matrix of instantaneous
transition rates.
1.4.2 Estimation Procedure
Zahl obtained the maximum likelihood estimates
and
provided the series converges elementwise, where
P*(t ' ) = P{t ' ) - I.
From (1.2) he also obtained for any t ~ t',...2 2 ...3 3
p( t) ::: I +At +A21 .. + A3 ~ •+
If
I nij :: 0, for some i,
j
(1,,3)
• A
P(t'} is not completely defined and hence, A andP(t) are undefined.
8
If the condition on (1.1) is satisfied so that ~(t') is completely
defined, the necessary and sufficient condition for convergence of
(1.2) is that the roots of the characteristic equation of P*(t') all
be less than one in absolute value.
1.5 Asymptotic Properties of the. Estimates
1.5.1 General Model
Since the Markov process model imposes the k linear restrictions
a :: l,2, ••• ,k ,
to avoid a singular distribution, it suffices to consider the joint
distribution of the t:0f31 13.# a == 1; 2, ••• ,k.
Zahl showed first that the set of estimates t0f3; a # 13,
(i) converge in probability to the corresponding elements of A;
(ii) are asymptotically jointly normally distributed with
covariance matrix equal to the inverse of a k(k-l) square matrix G
with elements
k k r-l
g0f3,13f == I ~(O)I I Pbi(at')b=l i=l a=O
OPij(t')
cn"al3
OPij(t')
dl.ysc,
(1.4)
where the partial derivatives are evaluated at the parameter values of
A;
(iii) in the limit as
t' -.0, rt' =T" constant"...
the asymptotic covariances of the ~0f3 are zero, so that, in view of
(ii), they are independent.
9
He then proved that the set of estimates Pij(t'), i ~ j, are
asymptotically linear combinations of the ~~ and are also jointly
normally distributed. The asymptotic means of the Pij(t') are the
corresponding elements of pet'), and their covariance matrix has the
elements
• . , ,_ \' \' dPij(t') OPum(t'). -1cov Pij(t )Pum(t ) - L L. oA· . 0)., gcq3,nr (1.5)
a;'(3 rh' ClJ3 rrr. -1 -1
where ~,nr elements of G , the inverse of (1.4) JI are, the asymptotic
covariances of the ~013' a: .; (3.
For a discrete time Markov process in which transitions are
considered to occur only at fixed time intervals, the parameter matrix
is of the form p( t ' ). Anderson and Goodman (1957) obtained, maximum
. likelihood estimates of p( t') for the discrete time model which are
equivalent to (1.1). TheY,showed properties of the estimates which, in
the notation ~sed here, are as follows:
(i) for each transient 8i , the set'ij(t'), J = 1,2, ... ,k, are
asymptotically normally distributed with meansPij(t D) and covariance
matrix
(1.6)
where k
Qi = Ib=l
r-l
I ~(O)Pbi(at')aDO
and 8jm~S the Kronecker delta;
(ii) the sets 'ij(t') and ~mj(t')" j == l,PHJlk, i 4: m, are
asymptotically independent.
10
l1sing these results for the ~ij(t') it can be shown, by a
proof analogous to Zahl's, that, conversely, the ~~, a ~ ~,are,
asymptotically linear functions of thepij(t'h and thus, jointly
normally distributed with means equal to the corresponding elements of
(1.7)
The covariance matrix of the t~ can then be obtained directly,
OAge OAzr Pij(t')[8 jm - Pim(t')]
OPij ( t • ) OPim( t ' ) Qi
A.
Proof that the matrix (1.7) is the inverse of the matrix (1.4) is given
in the appendix 7.1. C~nsiderab1y less computation is required to
evaluate (1.6) and (1.7) than (1.4) and (l.S) since no matrix inver-
sion is involved and the matrix of partial derivatives occurs in only
one of the equations. The matrix of derivatives
~op(t'}
may be computed by differentiating the series (1.2) in P*(t') with"
respect to pet'), and evaluating at the parameter values of pet').
1.5.2 Particular MOdel
In the particular model with only tWQ transient states the
summations in (1.4) and (1.7) are over i := 1,2, and j = 1,2,3,4. If
(1.4')
(1.8)
the individuals are initially in the same state, Sl' say, there is
simplification yielding
2 r-l 4
8afa,71T =NI I Pli(at') Ii=1 a=O oj=l
In (1.6) and (1.7) Qi becomesr-l
I \
Qi = N La...O
11
1.6 Merit of the Estimates in Follow-up Studies
Studies of long-term disease are generally severely limited in
sample size. The merit of Zahl's estimates in a Markov process model.
for such, cases must be Judged on the basis of small sample propertie,s.
Information was needed as to the rapidity with which the distribution
of the estimates approaches the limiting distribution as the sample
size increases, or alternatively as to the magnitude of the deviation
from the limiting distribution for fixed sample size.
Since the distribution of the t .. appeared to be mathematically1J
intractable, an investigation for the particular model was undertaken
by the Monte Carlo or empirical sampling method with computations per-
formed on the IBM 650. The results of the sampling for the smaller
sample sizes, N = 5, 10, were considered unsatisfactory because of the
high proporti~n of samples for which the estimates were undefined.
Evaluation of the probability generating function of the model for
sample sizes in this range was then undertaken to supplement the
sampling study.
Since A is the parameter matrix,of a continuous time Markov
process, primary interest in the application to long-term disease is
in the properties of the :S:ij' The matrix p(t) varies with the time
period considered. Thus the Pij (t I) per ,se give no information about
the model for, other time periods ~ However, inasmuch as th~, :tij are"
functions of the Pi.(t'), the properties of the PiJ.(t') are also of, J
interest .', Since the Pij (t ') are equivalent to the estimates obtained,,
by Anderson and Goodman (1957) for the discrete time model, results
obtained on their small sample properties are also of interest for that
model.
2. MONTE CARLO INVESTIGATION OF S~L SAMPLE PROPER.TIES OF
ZAHL ' SESTIMATES
_ 2.1 Scope and Method of sampling
For small sample size the properties of the estimates are, or
at least may be, dependent on:
(i) k, the number of states in the model and the proportion of
absorbing states;
(ii) the initial state of the members of the sample;
(iii) A, the parameter matrix;
(iv) t ' , the length of the interval between observations;
(v) r, the number of observations per individual. In order to
keep the scope of the investigation within reasonable bounds it was
necessary to limit the values of these factors for which samples were
Obtained. Since in application of the model t~ r, and N are the
factors over which some measure of control may be exercised, these
were given primary consideration.
Sampling was done with all individuals initially in the first
(transient) state, i.e.,. active disease. One matrix of values A for
the particular model described in 1.3 was selected for study. The
values of A were chosen with a view to obtaining a probability transi-
tion matrix p(t') for t' == 1.0, which would be meaningful for a severe
long-term disease, such as the chronic leukemias. These values of A
and P(l.O) are given in Table 7.4.1. For a process in which the
relative magnitude of the p•• (l.O) differed substantially from those1~. .
13
studied, ~he elements of the underlying A matrix would also differ,
but to a lesser degree. The use of a single A matrix somewhat limits
generalization of the results of the investigation. However, some
informa~ion was obtained on the effect of the magnitude of the Aij on
the properties of the estimates which indicates that the conclusions
hold for a fairly wide range of A values.
The major part of the sampling was done for t' =.1.0. In many
follow-up studies either information can only be obtained on patients
for a few time periods, or it is important to reach a decision regard
ing the relative efficacy of treatments after only a few time periods.
Analyses of data on 3 to 7 observations per patient are not uncommon.
For this reason samples for t'= 1.0 were generated for r = 3, 5, 7,
time periods. Properties of the estimates from samples over a greater
number of time periods would be expected to be at least as good.
Sample sizes of N = 5, 10, 20, 50, 100, 200 were selected as ranging
from the smallest encountered in practice to beyond the largest
probable under the most favorable conditions.
In a Markov process with one or more absorbing states it is
known that for given A as t increases the diagonal elements of P(t),
Pii(t), decrease and that the Pij(t), Sj absorbing, increase. In
order to gain some insight into the effect that the time period might
have on the properties of the estimates, samples were generated with
P(0.2) and P(2.0) for r = 7, and N = 10, 50, 200. These probability
transition matrices have diagonal values which are substantially
larger and smaller, respectively, than those of P(l.O) as may be seen
in Table 7.4.1. Values of the asymptotic covariance matrices of
Pij(t') and t ij are given in Tables 7.4.2 to 7.4.6.
14
Samples were obtained from pseudo-random numbers generated on
the IBM 650 using a statistical interpretive system program (Haynam,
1957). The r successive pseudo-random numbers interpreted under the
appropriate probability transition matrix p(t) constituted the transi-
tions by an individual. The sum of these over the N individuals yielded
the sample transition matrix [nijJ. From it the estimates Pij(t') were
computed to 6 decimal places and the matrix ~(t') tested for complete-
ness. If the sample failed the test it was coded as incomplete; other-
wise the test for convergence was applied. If the sample failed this
test it was coded as divergent. For convergent samples the ~ matrix
series was computed with termination when the bound on the remainder
term was less than 0.005 for all elements or the number of terms
exceeded 200.
For each combination of ~, r, and N, samples were obtained in
two separate sets of 50. This provided a pair of values of the
empirical means and covariance matrix of the estimates and the range
of the pairs gave a measure of the variation of the empirical sampling
method.
2.2 Deficiencies of the Estimates for Small Samples
2.2.1 Incomplete Samples
Under initial conditions that all individuals are in the same
state, SI' say, there is a positive probability ~f a sample such
that for i F 1, nij = 0, for all j, and the corresponding row of
P(t') is undefined. This type of sample will be called 'incomplete.'
For small rand N, the probability of an incomplete sample, which is a
15
function of pet'), may be of sufficient magnitude to cause appreciable
difficulty in applications. For the particular model investigated an
incomplete sample can obtain only if no individual enters 82 by the
r-l observation and hence, no transitions from $2 are observed.
The probability of this event is as follows. Let h(r,t'}
denote the probability that an individual does not enter 82 by the
r-l observation. Then, since individuals are assumed to behave
independently,
Pr(a sample of size N is incomplete) = [h(r"t,)]N 0
It may be seen from the model that an individual can only reach 82
from 81
, This transition is only possible if no transition from 81
to 83
or to 84
, the absorbing states, precedes it; that is, only if
the individual remains in 81 until the tran~ition to 820 Thus,
Pr(reaching 82 first time at bt') = [Pll(t,)]b-lp12 (t D), b =l,. •• ,1:''''l.
(2~1)
Now
(2.2)
= 1 -
h(r,t ' ) = 1 - Pr(reaching 82 at some btD, b = l, ••• ,r-l)
r-l
= 1 - L Pr (reaching 82 first time at bt')
b=l
r-lL. [Pll(t,)]b-lpl2(tl)
b=l
P12(t'){l -. [Pl.i(t,)]r-l).h(r,t
J) = 1 - ... l.,. P1l(tt) ...
For the general model with, say, m=l transient states other
than the initial state, m < k, an absence of transitions from anyone
or more of the m-l states results in an incomplete sample. The
16
probability that an individual does not reach anyone of these states
is then the complement of a sum of m-l terms. For any particular
model the probability can be found but for m > 2 the expression
becomes unwieldy.
Values of [h(r,t')JN were computed, using (2.2L for the A, .
matrix of the sampling and the t = 1.0, 0.2, and 200 for various
values of rand N. These are given in Table 7.4.7. From the values
for e = 1.0 it may be seen that the probability of an incomplete
sample decreases more rapidly with increase in N than with increase
in r. Alsoi
limr -+ (Xl
I P12(t ') JN
1 - Li -PH ( t ' ) > 0 i
whereas
It is intuitively obvious that if no transitions from a par-
ticularstate are observed in the sample there is no information on
the probability of those transitions for the given time period or on,
the instantaneous rates of the transitions. As indicated previouslYi
the proportions of samples for which estimates were undefined for
N = 5, 10 were so great that insufficient data were obtained to judge
the properties of the estimates. The frequencies of the types of
problem samples discussed in this section are given in Table 201 for
the combinations of t~ r i and N which were investigated. Comparison
of the proportion of incomplete samples to the probabilities of the
appendix table for the corresponding t' and r forN = 5, 10J 20 reveals
close ag~eement, improving with increasing N.
Table 2.1. Frequency of Monte Carlo samples by type and by t J ,r, N
17
t JN
r Type 5 10 20 50 100 200
0.2 7 Incomplete 20
Divergent 3
Convergent 77 100 100
Negative A, 30 11 71
1.0 3 Incomplete 35 18 3,
Divergent 21 23 8 2
Convergent 44 59 89 98 100 100
Negative A, 22 30 66 84 72 50- --
S Incomplete 23 4
Divet'gent 17 15 6
Convergent 60 81 94 100 100 100
Negative A, 44 64 78 62 49 42
7 Incomplete 19 2
Divergent 26 6 1
Convergent 55 92 99 100 100 100
Negative l 42 81 76 55 54 48,
2.6 7 Incomplete 6
Divergent 33 2
Convergent 61 98 100
Negative ~ 46 56 45I\,
18
2.2.2 Divergent Samples
~ is defined by (1.2) only for those complete t(t') yielding a
convergent series in ~*(t'), the necessary and sufficient condition
being that the roots of the characteristic equation of t*(t') all be
less than one in absolute value. This condition is not convenient to
apply in practice. Zahlshowed that a sufficient condition is that
the Pii(t') be greater than 1/2 for all i, and s~gsested that, on the
basis of prior information, the time interval of observation for the
process be chosen small enough that the expected proportion of transi
tions from any state to some different state be less than 1/2. Since
thepij(t') tend in probability to the true values of Pij(t') with
increasing sample si~e, the probability of a sample which did not
satisfy the sufficient condition would then be very small for 'large'
samples. However, it is questionable whether prior information would
always be available to enable such a judicious choice of time interval.
Even assuming that it were, the interest here is in cases where the
sample size is so small that the probability of a sample not satisfying
the sufficient condition would still be appreciable.
No convenient necessary and sufficient condition for the
general model has been found, but given a sample matrix P*(t') for any
k state model, it is possible to determine from the numerical values
whether the condition on the roots of the characteristic equation is
satisfied without finding the roots. Let the transitions from the
transient states occupy the first m rows of t*(t'). The characteristic
equation,
19
may then be written as a polynomial of order m in e,
m m-l m-2 me - f3 l e + f3 2e - ... + (-1) 13m :: 0,
where f3i is the sum of the i th order principal minors of the submatrix
formed from the first m rows and columns of ~*(tJ). Samuelson (1941)
described a method of obtaining necessary and sufficient conditions on
the coefficients of a polynomial such that the roots be less than
unity in absolute value. This method can readily be applied to
numerical coefficients but for a general equation greater than the
third order it leads to very complicated expressions. Hence, in
using (1.2) for any k state Markov process with three or more transi-
ent states the following is suggested:
(i) any prior information on rate of transitions between the $t._tes
be used to choose a time interval between observations small enough
that the expected propc:ii·tion of transitions from any state to the
same state be substantially greater than 1/2;
(ii) the sample matrix ~(t') firs.t be examined for the sufficient
condition
Pii(t')> 1/2, for all i,
and if this is satisfied, the ~ series is defined;
(iii) if not, the coefficients defined above for the polynomial in
e be evaluated from the rows and columns of !*(t') corresponding to
the transient states and the method of Samuelson be used to determine
if the roots are less than one in absolute value so that the ! series
is defined.
For the particular model the conditions·on the coefficients of
the characteristic equation yield the condition
20
(2.3 )
. which is equivalent to
IP(t')1 > 0 .
Since the sign of this quantity determines whether or not the sample
yields a convergep.tt. series, it was conjectured that the probability
of a divergent sample may be a function of the determinant of the
parameter matrix P(t').
The values of the determinant of p(t') for t' = 0.2, 1.0, 2.0
are 0.825, 0.383 and, 0.147, respectively. It may be seen from
Table 2.1 that the frequencies of divergent samples for r =7 and
N = 10 for these t' are in reverse order to that of the parameter
determinant values.
'The meaning in physical terms of a divergent sample is more
difficult to see than that of an incomplete sample. For the particular
model a divergent sample Qbtains when the product of the transitions
The same inequality holds for the estimates from the sample,
It can be shown for the probability transition function matrix of the
particular model that
•
21
Thus a divergent sample provides. a set of estim~tes which is invalid
for any probability transition function matrix of the model.
2.2.3 Finite Termination of the ~ Series;
If a sample obtains which yields both a complete ~(t') and a
converge~t series (1.2) in ~*(t'), there is still the question of the
magnitude of the error in terminating the series at a finite number
of terms. Since t is a power series in a matrix argument, the error
term is also a matrix. The magnitude of the error can thus be defined
in terms of either the absolute value of the determinant of the error
matrix or of the maximum absolute value of the elements. Since com-
parison of the properties of 'the elements of the Aestimate matrix
was of interest in the Monte Carlo investigation, termination of the
series computations was programmed in terms of a bound.on the maximum
absolute value of the elements of the error matrix. This bound holds
only for the particular model studied since it depends on the form of
[~*(t')]S for the model.
Since the bound is for a matrix of numerical values, the time
dependent and estimation notation is omitted here. For this model the
pattern of convergence is such that, after the first mo' say, terms,
eaChP:}s) has alternating sign fo~ successive s and decreases in
absolute value, unless identically equal to zero. Hence, for every,
ij, Aij is a sum of mo terms plus a convergent infinite series of terms;
of like sign. The finite term,mo' beyond which the bounds on the
remainde.r hold, is determined by the conditions
j = 1,2,3,4;
(ii) all i,J.
22
For m > m , denote the remainder matrix of (1.2) after m terms by. 0
and let
() max ()v m . &I i,j v ij m •
v(m) Ip*(m-l) I< ij ..
1 -v(m) 1
Then
&I 0, *(m-l)Pij = 0 ,
The derivation of the bounds is given in appendix 7.2.
2.2.4 Negativetij Values
A further difficulty is encountered in the use of t series
estimates. A sample may obtaip, in which one or more Pij(t'), i .;. j,
is zero when the corresponding parameter value is greater than zero.
The ~matri~ obtained from the series in such a ,*(t~) may then contain
one or more non-diagonal terms with negative values. These are
invalid estimates since the Aij , i F j, are non-negative by definition.
If such a sample obtains in practice, the experimenter is faced.with
the problem of making the best use of the only available information,
a negative value which is meaningless for the assumed model.
Since the exact distributional theory of the estimates is lack-
ing for finite sample size, a solution to the problem can be proposed
only on intuitive grounds. The probability o~ a sample yielding a
negative estimate would be expected to increase as the· value of the
parameter approached zero. Thus from a negative estimate it would
seem reasonable to infer that the parameter is near zero and redefine
the estimate as zero for negative valu~s. The modification of nega
tive t ij for one or more j # i necessitates adjustment of other
tbelements of the i row to preserve the defined linear relationship
It may be noted that the probabilities of transition in an infinitesi-
mal time period /;)t. are
Pij(At) = AijAt -1' o(at), j rf: i,
PU(At) = 1 + AUAt + o(At),
where AU =. -I Aij •j,&l
If Aii is modified by -E, say, the proportional change inPii(At) is
-eAt
-eAtAikAt
For this reason it was considered better to compensate for the modifi-
cation of negative t ij by adding the negative value to the diagonal
. estimate tii
•
It is recognized that the modification adopted for investigation
in this study is not the only one that might be proposed and that it
may, in some sense, not be the best. It was chosen as being reasonable,
simple, and workable. The f11,odified t ij are not likely to be maximum
24
likeliho04 estimates under the restraint that A23 = 0, and their distri
bution would be expected to approach the asymptotic more slowly. The~ij
were obtained in both modified and unmodified form for all convergent
samples and differences in the d~ibuti6na are presented in section 4.
2.2.5. Discontinuity ,of Pij(t') and the Effective Sample Sample Size
For finite sample size the Pij(t') are discontinuous statistics,
taking only values which are integral multiples of l/ni •• The variable
n. may be designated the effective sample size with respect to1.
Pij(t'), j =l, ••• ,k. For the particular model sampled the values of
,n2• were alw~ys less than N and the frequency distributions for N = 5,
10, were very skewed toward zero. Such values of n2• cannot provide
precise estimates of small P2j(t'), much less of the ~ij which are
functions of them. Hence for small sample size the distribution of the
effective sample sizes, nL
, has an important role in the distribution
'of the estimates.
all individuals in the same state initially, say S1'
N<n' <N,1. r
6 < n1 • < N(r-l), i # 1.
Thus the range of ni ., i # 1, may overlap that of nl • but is
.25
essentially to the left of it, since the lower and upper limits are
less tl1an those o·f n1•• It can be shown that the mean values of the
ni • are given by the Qi of (1.8) which may be written
r-l
Qi :.: N(81i + I P1i(at')] ,
a=l
where eli is the Kronecker delta. From Markov process theory it is
known that for S. transient, Pi,(at') decreases with increasing a.J J
Consequently, for PU(t') approximately equal to, or greater than,
Pli(t'), Qi will exceed Qi, i ~ 1, by approximately N.
The n. are sums of identically and independently distributed1.
variables, the transitions by the individuals of the sample. Hence
the moments of the ni. for a sample of size N are the N-fold Il)ultiples
of the moments for a single individual. The latter were evaluated
from the probability generating function and are discussed in section
3. Further, the marginal distributions of the ni. will approach the
normal with increasing N.
2.3 Definition of Conditional Probability Distributions
of Pij(t'), t ij
--The frequency of incomplete and divergent samples for N = 5,
10, 20, indicates that for small N the subsets of the sample space of
the (nij ) for which P2j (t ') and/or t ij are undefined have probabilities
substantially greater than zero. The question thus arises of the
definition of valid probability distributions for such statistics,
Conditional distributions, as formulated below in general terms, were
used in studying the Pij(t') and t ij ,
26
Let ~ be a discrete k-dimensional vector variable with
probability function NPv defined over the space ~ and the set
function
so that
Consider a one-to-one function ~(r-~J such that
~ is { defined for If- E: Jil' C -Jl '
undefined for ~ € ~ - Jil' .
Denote bfJ(r#:v) by Jli'v and for any set S C If- I define
S ;: L,j;) : _.x € S] •e l'rv~-v
Then
is the space over which the function 119 is defined. Now
and
But if NPv > 0 for SO!l1e N!v € -Jl - ~ I, then PN(~') < 1, so that NPv
is not a valid probability function for NS'Let 'N = P(~'), and define the conditional probability function
of II'
27
Then
G( n) =N .N e = 1,
so that NSv is a valid probability .function for Ne. Further, if
lim... = 1n 00+ 00 'l'N '
NSv becomes the unconditional probability function. The conditional
mean is
= .!...ljIN
and the conditional variance is
2NIJ" g =
Hence, in the limit, as N 00+00, the conditional mean and variance
become ~he unconditional mean and variance.
The conditional means and variances of the Pij(t') and the t ij
were estimated frpm the unconditional means and variances of the sets
of samples which yielded estimates.
3. EVALUATION OF THE PROBABILITY GENERATING FUNCTION
3.1 Discussion
The high frequency of incomplete and divergent samples of size 5
and 10 in the Monte Carlo investigation resulted in too few convergent
samples to yield reliable information on the properties of the esti
mates t and of the convergent ~(tJ) of which they were a function.
Rather than generate the large num~er of additional samples necessary
to provide an adequate number of convergent samples, the probability
generating function was evaluated for N ~ 5.
No report was found in the literature of prior use of computer
/ evaluation of generating functions to study sampling distributions.
For this reason some details of the method of evaluation and the
difficulties encountered are presented in appendix 7.3.
Information on the sampling distribution for the smallest sizes
was obtained from the generating function, and this portion of the
investigation is discussed before presenting the Monte Carlo results.
3.2 Definition of the Probability Generating Function
For the particular four state model, transitions by an indi-
vidual are governed by the probability transition function matrix
PH (t) 1'12(t) P13(t) P14(t)
p(t) "'\P21 (t) P22(t) P23 (t) P24(t) ..
=0 0 1 0
0 0 0 1
29
The probability generating function of the transitions, {nij }, by a
single individual during r time periods of length t, given that he was
initially in Sl' is .
\' nllk n12k n14k n2lk n24kGr[1,8ij ;P(t)] =L akSll 812 ...814 e2l ,,,S24 ,
k
where the coefficient ak is the probability that the vector (nij )
takes the values (nijk), and the summation is over the set of vector
values (nijk) admissible under the model. The probability of a vector
value (nijk) has the form
nllk nl4k n2lk n24kak =: '1tPll(t) ...P14(t) P2l(t) ...P24(t) ,
where ~. is the number of paths an individual may traverse leading to
. the same vector of transitions.
Since individuals are assumed independent of one another in
their transitions, the generating function for N individuals is
3.3 Distributions Obtained from Probability Generating Function
Evaluation
For t =: 1.0, r "" 3, and sample sizes 1 to 5 the evaluation
yielded~
(i) the probability distribution of the set of admissible vector
values (nijk),
(ii) the joint probability distribution of nl.' n2.,
(iii) the joint probability distribution of (Pij ) by type of sample,
convergent, divergent, and incomplete,
30
(iv) the marginal distributions~ conditional mean vectors and
covariance matrices of the (Pij ) for convergent and complete samples,
and of the (PljJ for all samples.
3.4 Sampling Distributions of the Estimates (Pij(l.O)}
for r :: 3, N 5. 5
3.4.1 Probabilities of Types of Samples
The distribution of the total prnbability over the spaces of the
three types of samples obtained from the evaluation of the generating
function is shown in Figure 3.1. The probability of an incomplete
sample, computed as described in section 2.2.1, is included for N to
30. Values obtained by the two methods. agreed within the limits of
computational accuracy. The Monte Carlo proportions of sample types
for N =5, 10, 20, 50 are included together with 95 percent binomial
confidence limits from the tables of Mainland, !! !l., (1956) to
relate the information obtained by the two methods of investigation.
An attempt was made to extrapolate the curve of the probability-
of a convergent sample for N 5. 5 by fitting the first five terms of
several power series in N and lIN. The fit for larger sample size was
judged by the Honte Carlo confidence intervals on the proportion of
convergent samples for N "'" 10, 20, 50. The power series approximations
fell outside the intervals with one series underestimating and two
series overestimating the probabilities.
3.4 ..2 Distributi9J1s of Effective Sample Sizes
Coefficients of the probability generating functions of nl .,
31
I.L
T95 010 confidence limitstt about the Monte Carlo.L complete proportion
T95 010 confidence limitso about the Monte Carlo1 convergent proportion
Ir
1IoIII1
+-Convergent sample(exact for N~ 5,Monte Carlo for N>5)
IIII
QIIIII
.L
Completesample(exact)
TI
Probability1.0 __------=---=------~r_--
0.8
0.9
0.5
0.6
0.7
0.4
0.3
0.2
0.1
O------r'---,..---..,....----,------.------t
Figure 3.1
10 20 30 40 50Sample Size - N
Probability of types of samples fort' equal 1.0, r equal 3, by sample size
60
32
were enumerated by the combination of terms of G [l,e.j,p(l.O)] forr J.
r =3, 5, 1, and the means and covariance matrices thus obtained.
These are given in Table 7.5.1.
were obtained for r = 3, N S 5.
The joint distributions of the n.J..
It is the trend of the distributions which is of interest as
indication of the approach to normality that may be expected as N
increases beyond five. Hence, only the distributions for N = 2, 5,
are presented in Tables 3.1 and 3.2 together with the MOnte Carlo
results for N = 5, Table 3.3. The range of the variables is, of
course, limited to the triangle
For N = 5, the probability surface is rather diffuse, but it can be
seen that for both N = 2 and 5 the larger probabilities are clustered
below the mean of n2• and roughly symmetrically about the mean of nl ••
The contours of approximately equal probabilities are very irregular.
The Monte Carlo relative frequency distribution of Table 3.3 is quite
similar to the probability distribution for N = 5.
Some approach to normality from N equal 2 to 5 can be seen in
the marginal distributions of nl • and nZ• shown in Figure 3.2. For
nl • the trend is marked. Not only is the distribution shifted sub
stantially to the right, but it is also changed from very peaked and
one-tailed to almost symmetrical. The change in the J-shaped marginal
distribution of n2• is of degree rather than shape. The range is
increased and the predominant probability of zero value reduced with
33
Table3.l. Joint probability distribution of effective sample sizes
nl •, n2•, from probability g~nerating function evaluation
for t equal 1.0, r equal 3, N equal 2
~0 1 2 3 4 Marginal
n1. probability
2 .039 .005 .024 .001 .003 .072
3 .055 ,.043 .019 .021 .129
4 .215 .039 .068 .322
5 .137 .098 .235
6 .242 .242i
Marginal .688 .185 .,111 .013 .003 1.000probability
34
Table 3.2. Joint probability distribution of effective sample sizes
n1., n2., from probability generating function evaluation
for t equal 1.0, r equal 3, N equalS
~0 1 2 3 4 5 6 7 8 9 Marginal
nl.prob.
5 a a a a a a a a a a a
6 .001 .001 .002 .001 .001 a a a a .006
7 .005 .004 .007 .003 .003 .001 .001 a a .024
8 .012 .012 .013 .010 .005 .003 .001 a .056
9 .030 .022 .033 .014 .011 .002 ·.001 .113
10 .045 .045 .038 .024 .007 .003 .162
11 .076 .050 .056 .016 .009 .207
12 .072 .065 .0.32 .01T .186
13 .081 .036 .029 .146
14 .041 .029 .070
15 .029 .029"Marginal
.999+prob. .392 .264. .210· .085 .036 .009 .003 a a a
aprobability less than .0005
35
Table 3.3. Joint relative frequency distribution of effective sample
sizes nl ., u 2., from 100 MOnte Carlo samples fort equal
.1.0, r equal 3 11 N equal 5·
-
n2• 0 1 2 3 4 5 Marginaln1- frequency
5 .01 .01
6
7 .01 .01 .01 .03
8 .02 001 .03 .01 .07
9 .02 .02 .04 .02 .01 .01 .12
10 .05 .03 .05 .01 .01 .15
11 .07 .03 .06 .04 .20
12 .09 009 .03 .21
13 .06 005 007 .18
14 .01 .01
15 .02 .02
Marginal.35 .22 .25 .14 .01 .03 1.00frequeney
36
Probabili ty.7
Probability.7
.6
--- Nequal 2- Nequal5
o Monte Carlo, N equal 5
.5
.6
.3 f,I \I ,II \_,
I '.2 I \, \, \, II ,
.1 I \I \
I I/ \
/I
02 ·4 6 8 10 12 14 16
nl·
Figure 3.2. Marginal probability distributions ofeffective sample sizes, n,., n2., fort' equal 1.0, r equal 3, N equal 2,5
37
an increase in each of the probabilities of the non-zero values. The
resulting distribution bears some resemblance to one-half of a normal
distribution. It appears reasonable to expeyt that the marginal dis-
tribution of nl for N equal 10 or 20 would approximate the normal, .
distribution. The s~mple size at which the marginal distribution of
nz. may become symmetrical and tend to the normal is problem~tical,
but it must be at least large enough that the probability in the left
tail, essentially that of zero values, is very small. From Table 7.4.7
this wo~ld be in the neighborhood of 30.
The Monte Carlo marginal frequencies in Figure 3.2 follow the
marginal probability distribution rather closely. This closeness may
be judged by the Kolmogorov-Smirnov statistic which yields values .07
and .09 for nL and n2•• From Birnbaum's tables (1957.) of the exact
distribution, the probability of a deviation this large or larger for
100 samples is 0.47. Thus, the Monte Carlo results yielded an adequate
representation of the sampling distribution of the effective sample
sizes.
3.4.3 Marginal Unconditional Distributions of (p.. (1.0)}1J .
The cumulative marginal distributions of the Pij show some trend
from sample size 2 to 5. The unconditional cumu~ative marginal distri-
but ions provide a comparison of the behavior of the estimates for the
nested sample spaces of convergent, complete and all samples. This is
the information relevant to the continuous time model, the discrete
time model, and transitions from the initial state of the discrete
model, respectively. For the subspaces the mean and variance of the
unconditional distributions are undefined, as was discussed in
38
section 2.3. The conditional parameters are presented in the follow-
ing sections.
Examples of the unconditional marginal distributions are shown
in Figures 3.3 to 3.5. Due to the fact that a c~mplete sample requires
non-zeroP12 the distributions of this variate for the complete and
convergent spaces differ from those of the other variates. The dis-
tributions of P14 closely resemble those of P13; t~ose of P2jare very
similar for all j.
It is apparent that the unconditional marginal distributions
over the subspaces are non-normal, those of P2J being more markedly so.
The latter is not surprising in view of the findings on the range and
distribution of nZ.' These estimate,s take a very limited number of
values with a predominance of zeros. There is little change in the
shape of the distributions up to N equal 5 over either the convergent
or complete space. Similarly for fixed N there is but slight
difference in the shape of distributions over the two spaces.
The cumulative distributions of the Plj
have a flattened
S-shape with irregularity in the lower tail. The lower tail of the
distributions of 1>12 over the convergent and complete spaces is absent
due to the restriction to Rpn-zero values •. As N increases the number
of values which the,Plj can take with appreciable probabi,lity increases
so that the S-shape of the distribution becomes less flattened and the
lower limit of the range of P12 tends toward zero. The greatest
approach to normality is seen in the change in the lower tails of the
distributions. The amount of change is not appreciably different over
the three sample spaces. However for fixed N the c~ulative distribu
tions of the Plj over the sp~ce of all samples resemble that of the
39
Probability1.0
0.9
0.7
0.6
0.5
O~
0.3
e 0.2
0.1
Complete samples
N·5
N·4
N-3
N-2
1.0.9.8.74 .5 .6
"Pl2 (1.0)
.3.2.Io
0 .1 .2 .3 ~ .5 .6 "] .8 .9 1.0Pl2 (1.0)
Convergent samples
04 Vl2 (1.0)N·5
N·4
0.3
N·3
0.2
N-2
0.1
Figure 3.3: Unconditional marginal cumulative probability distributions ofPl2 (1.0) for r equal 3, by sample size and by type of sample
40
N-5
N-4
N=3
N-;:2 --r--------
Complete samples
0.1
0.7
0.8
0.9
0.6
0.3
All samples
~r-~~-~-'-' --'--'-'-'-,,::-~='=='-=--":"'-':=-::'-='=J_r'-' N- 2 I
f r----~--II-'-'~
i Ir-.1 I
J r----_.J ,---
I I._1 I!.- J
rr-iI I
,.,..i I. II I
r.J Ii Ii ,.._.Ji I
'" I! I! I
r.J,..J! Iiii
----ri.F~I..JO~
I,..'
r I! I
0.2~--j"I.,::I--.J:---I
~I
0.5
Probability1.0
0 .I .2 .3 4 .5 .6 .7 .8 .9 1.0" 0.0)Probability Pl3
Convergent samples N=5O~
N-4
0.3
N=3
0.2N=2
0.1Pl3 (1.0)
0 .1 .2 .3 ~ .5 .6 .7 B 9 1.0• (1.0)Pl3
Figure 3.4. Unconditional marginal cumulative probability distributions ofPIS (1.0) for r equal 3, by sample size and by type of sample
Probability Complete samples
0.6 L -r---r--N~-~5-----I"---------------J
N-40.5W~ --r--...r--~~-..r---------_--J
N-3
O~
N-20.3-1-.:.-..__----....r--:....:..-=--~---------------J
41
0.2
0.1
0 .I .2 .3 ~ .5 .6 ? .8 .9 1.0
P23 (1.0)
Convergent samples
O~N=5
N-4
0.3
N-3
0.2-
N-2
0.1 ~I
0 .I .2 .3 ~ .5 .6 .7 .8 .9 1.0A (1.0)P23
Figure 3..5 Unconditional marginal cumulative probability distributions of" (1.0) for r equal 3, by sample size and type of sampleP23
42
normal to a greater degree than do the distributions over the space of
complete samples.
It should be noted that the distributions of the Pij over all
three spaces for N equal 5 are asymmetric about the parameter values
of which they are estimates. The asymmetry of the small sample dis
tribution of multinomial estimates is well known for values of the
parameter in either /extreme of the range and the;Pij differ from
multinomial estimates onl~ in having a variable rather than constant
denominator. The parameter values Pij(t) of the mOdel will, in
general, be small so that the aSY!JUlletry of the estimates is of real
concern.
3.4.4 Conditional Bias by Sample Type
Data on the bias and percent bias of the Pij from the conditi~
distributions for the three types of samples are given in Table 7.5.2.
The trends with increasins N are more easily compared in terms of the
percent bias. These are shown for Pij in Figure 3.6 and for P2j in
Figure 3.7~ It can be shown mathematically' that the percent bias of
the estimates PlA(t)Is .constant for all SA abso:rping,a.nd similarly
for P2A(t). Hence, one set of curves is included in each figure for
. the estimates of transitions to absorbing states. In section 4
-1 -2regression functions ofN for the bias and N for variances are
fitted to the Monte Carlo data. For the sake of consistency and com-
parison, bias and variance data of this section are plotted against
-1N • The magnitudes of the percent bias for N equal 1 were so great
relative to those for N from 2 to 5 that they added little to ~he trend
and for convenience of graphing were not included.
43
Percent160
140
120
100
80
60
-- PI2 (1.0)
--- -- PIA( 1.0)"Convergent samples
'-Complete samples
IN
_--- All samples-----------o 1-- ----.;.,.Ii--_--.:..:r~_ ____:...r_--.-;4_r__--.5r_~----.... .......
""...... -- ....-, --, .... - Complete samples, ---........ ....-......
" " "" '"Convergent samples "",
"
20
-20
-40
Figure 3.6. Percent bias of Pl2 (1.0) and PIA (1.0), SA anyabsorbing state, for r equal 3, by reciproca Iof sample size, by type of sample
Percent60
40
-20
-40
-60
-80
-100
"-- P21 (1.0)
"----- P2A (1.0)
................ Convergent samples-- ............
.... _- ------
44
Figure 3.7. Percent bias of P21 (1,0) and P2A (1.0), SA anyabsorbing state, for r equal 3, by reciprocal
of sample size, by type of sample
45
The percent bias of PlZ is positive and substantial for both
convergent and complete samples. The curves cross so that only from
N equal 4 is the bias less for complete than for convergent samples"
as would be expected for the larger sample space. The positive nature
of the bias might be attributed to the exclusion of zero values from
the conditional distributions. However" the bias for all samples is
also positive although of substantially lesser magnitude~
The bias of PIA on the other hand is negative for both corr~
vergent and complete spaces. The curves for both types of samples are
similar. The rate of change of the curve for convergent samples is
greater so that the two curves are approaching with increasing sample
size, as would be necessary as the probability of divergent samples
approa'ches zero and the two sample spaces become the same. The
direction of the bias of PIA for all samples is reversed to positive
and the curve of percent bias is very similar to that of PIZ '
The bias of both PZl and P2A among convergent samples is
negative like that of PIA' The percent bias OfPZl is greater than
that .of PZA but decreasing more rapidly so that it appears to be
approaching it. Among complete samples the bias of both estimates is
positive in contrast to that of PlA' The values of the percent bias
of P2l and P2A were equal to the accuracy of the computations. No
theOretical basis for this equality could be found and hence, it cannot
be concluded that it holds in general.
For both convergent and complete spaces the magnitudes of the
A d A Apercent bias of P2l an P2A for N > 2 are greater than those of PIA'
Moreover" the rate of decrease over the range evaluated here is very
46
slow, particularly for complete samples, so 'that for "larger N the
bias of the P2' cannot be expected to decrease very rapidly.J . ,
3.4.5 . Conditional Covariance Matrices by Sample Type
The conditional covariance matrices of the Pij , j ~ i, for the
three types o~ samples are given in Tables 7.5.3 to 7.5.5. Figures
3.8 to 3.12 show representative variances and covariances plotted
-1against N for the three types of samples. The vertical scale units
are inversely proportional to the magnitude of the asymptotic vari-
ances and covariances. Thus in comparing two figures, equal deviations
-1from the asymptotic lines at a given value of N represent equal per-
centage deviations from the asymptotic variance or covariance. The
Monte Carlo values obtained from two sets of samples are included for
both complete and convergent samples of size 5 and 10. One set of
samples of size 5 yielded positive values of the covariance of P13 1P14
of .001 for convergent samples, and .003 for complete samples. These
values were outside the range of the graph.
The variance curves of the Pi' for all three types of samples. J
appear to be approaching each other and the line of asymptotic
variance. By N = 5, the variance curve for all samples of the Plj
is
farthest from the asymptotic, that for complete samples closer and for
convergent samples closest. For the P2j on the other hand, the com
plete variance curve is much closer to the asymptotic line than the
convergent curve. However, since the complete curve is approaching
the asymptotic line from below it is possible that for N > 5, it would
cross the line and then approach it from above, in the same manner as
the convergent curve of the Pljo
47
II
I'I
//
all samples
Monte Carlo estimateso convergent samples6. complete samples
/. comp lete //
samples ~_ ...I /~ asymptotic
6 .I //./ ///
/// ----o . / convergent samples%/y
/g)'
.01
.03
04
.02
Variance.05
o-------,..--~---.,..--......,---......-+.I .2 .3 4 .5
IN
Figure 3.8. Variance of. Pl3 (1.0) for r equal 3, byreciprocal of sample size, by. type of sample
48
Variance.08
/ asymptotic
/l!. / complete samples
/-
6 _--------...,..,-----
.07
.06
.05
.04
.03
Monte Carlo estimateso convergent samples6 complete samples /
/
IN
.54.3.2
oconvergent samples
.10 ......--.......----4ir---..,.---.....,....----,-+
.01
.02
Figure 3.9. Variance of P23 (1.0) for r equal 3, byreciprocal of sample size, by type of sample
49
Covariance.
.002
.001
o
//
//
// complete samples
//
//
//
//
/ convergent samples/
//
/
Monte Carlo estimateso convergent samples6 complete samples
o
o
.1
"'"~: ""
\~
.\.~
\ ~
\ ""\ ~ asymptotic
\ ~
o--IP---~-----r----,..--__~----,r---+ IN
-.003
-DOl
-002
-004
Figure 3. 10.
\ all samples
Covariance of Pl2 (1.0), Pl3 (1.0) forr equal 3, by reciprocal of samplesize, by type of sample
50
Covariance.0005
.0036t.001 0 out of range
.\ all samples
. convergent samples
\
Monte Carlo estimateso convergent samples
6 complete samples
o
complete samples
6o
O-l---...,....---r----.-'"I""""""---r---or--.... IN
-.0005
-.0015
-.0010
-.0025
-.0020
Figure 3. II. Covariance of Pl3 (1.0), PI4 (1.0) forr equal 3, by reciprocal of sample
size, by type of sample
51
Covariance.5.3.I .2 I
O-'----'---~--......--......--........~Nconvergent samples
-.005
Monte Carlo estimates
o convergent samples6 complete samples
-.010
-.015
-.020
-025
6_"""~ - '\..- _ complete samples'"------~ ---
~~SYmPtolic~
"
Figure 3.12. Covariance of P21 (1.0), P23 (1.0) for
r equal 3, by reciprocal of sample size,
by type of sample
52
In the covariances of Plj'Plk, the curve of all samples appears
to be approaching the asymptotic in a regular manner and by N = 5 is
quite close to it. The convergent and complete curves approach the
line at an angle which would result in intersection so that a reverse
curvature for N > 5 would be expected.
Curves of the covariance of P2j,P2k" also approach the asymptotic
line at abrupt angles and the complete curve has in fact crossed the
line by N = 5, departing from it. This would also be expected to
change direction for N > 5.
The Monte Carlo values for N = 5 appear consistent with the
generating function"values of the variances in that either both values
are very close to the curve or the pair has a range which brackets the
curve. For N = 10 the Monte Carlo values appear consistent with the
expected direction of the curves. The pairs of values of the covari
ances vary so widely "in most cases that it is not possible to draw
conclusions about their relation to the generating function curves.
3.4.6. Conditional Mean Product Error Matrices
Since many statisticians hold the mean square error to be the
most important single measure of an estimator, for completeness the
values of the mean square errors and mean product errors of the Pijare included in Tables 7.5.6 to 7.5.8. The general trends with sample
size may be observed from the tables.
4. RESULTS OF MONTE CARLO INVESTIGATION
4.1 Choice of Methods of Analysis of Resultsi
Each of the sets of estimates (1.1) and (1.2) has an asymptotic
multivariate distribution. A multivariate test is therefore required
to determine whether the distribution for a particular sample size is
satisfactorily approximated by the asymptotic distri~ution. Also, a
test which is distribution-free would be needed to test both porma~ity
and the parameter values. A multivariate distribution-free test is not
yet available (Rosenblatt, 1962). In cont1.ast to the univariate case,
deviations of the sample distribution function of a vector variable are
not independent of the distribution of the variable (Simpson, 1951).
Thus univariate tests based on this statistic do not extend to the
multivariate case.
If one assumes normality of the joint sampling distribution for
finite H, the standard normal theory test of the parameters can be
applied. However, in the multivariate case the class of alternative
hypo~heses to the simple null hypothesis is very wide; rejection of
the null hypothesis implies only that the asymptotic values do not
hold for at least some of the parameters. In addition to determining
the sample size for which the asymptotic distribution of the set of
estimates is acceptable it was desired to determine, for the smaller
sample sizes, which paramaters differ from the asymptotic values and
to obtain some measure of the deviation. Hence, it was decided that,
in the event of rejection of the specified multivariate dis'tribution;
univariate tests of the marginal distributions and univariate methods
54
of estimation over the range of sample size would be used to draw
inferences about properties of the small sample distribution of the
estima~es. It is recognized that a substantial amount of correlation
exists between all the tests used aud also between the results for the
.{~ij(t ')) and the (tij ): While in general such a procedure in statis
tlc~l aualysi~ ise.subject to criticism, it is felt that inanexplora
tory study of this type it is preferable to obtain more definitive .
information with a lesser degree of reliability. Such information may
point the way to further research.
In analyzing the marginal distributions by univariate methods
the Kolmogorov-Smirnov test of a completely specified distribution
permits inclusion of normality in the null hypothesis. Rejection of
the test can be due to deviations from normality and/or from the
parameters. The univariate tests of normality proposed by Geary
(1947) are known (Kac !! !!'j 1955) to be instead tests of skewness
and kurtosis, since many non-normal distributions exist with properties
of the third and fourth moment about the mean equivalent to those of
the normal. The Kolmogorov-Smirnov test was chosen over the chi-square
test for greater computational ~onvenience to avoid the problems of
selection of number of classes and class intervals, and for its
generally greater power.
Both the multivariate test of the mean vector and covariance
matrix and the Kolmogorov-Smirnov tests of the marginal distributions
were applied to the individual sets of data by sample size. These were
augmented by univariate analysis of the data on means and variances
(covariances) over the investigated range of sample size" Since
asymptotically the statistics are unbiased" with variances (covariances)
55
which are functions of the reciprocal of sample size, it was assumed
that the bias is of the order of N- l and that the variances (covari
-2ances) are the sum of the asymptotic value plus a multiple of N •
1 -2The null hypotheses of zero coefficients of N- and N were thus
equivalent to hypotheses of the asymptotic means and variances
(covariances), respectively. In addition to providing estimates of
those parameters for which the asymptotic values were not acceptable,
the regression tests were more specific for bias and variance
(covariance) than the marginal frequency tests.
4.2 Multivariate Analysis
4.2.1 Mean Vector and Covariance MatriJ!,
4.2.1.1 Description of Test. For any parameter point (t')r,N),
given W6-component observation vectors on the (Pij(t'», i == 1,2,
j == 1, •• 0,4, and i # j, (or on the (~ij» which are assumed to be
jointly normally distributed with mean vector Ii and covariance matrix
n, the null hypothesis to be tested is
H:fJ.==Ii,fi==Q000
In the case of (Pij(t'»,
fJ. == (p~j(t'», i == 1,2, j == 1, ••• ,4, i # jop &
where 1(1' te 2, are 3 X 3 with elements given by (1.6) with denominators
Qi of (1.8). In the case of (~ij)'
== i i= j,
and the elements of fioA are given by (1.7).
• 56
The likelihood criteriane (Anderson, 1958) is described irt the
appendix 7.7.1. The approximate chi-square distribution of -2 log ee
with 27 degrees of freedom was used in testing the hypothesis. This
test was applied only to the convergent samples.
4.2.1.2 Results. The test was applied to each of the two sets
of 50 or less convergent samples for a parameter point. Since the
sets of samples were independent and the test statistic distributed
as chi-square, the sum of the two test values was distributed as chi-
square with 54 degrees of freedom. The data on (Pij(t')) were tested
for all ~ and r for successively smaller N from 50. The pairs of
values of the test statistic and their sums are given in Table 4.1 with
significance at the .05, .01, and .001 level designated by single,
double and triple asterisks, respectively. It is concluded that for N
equal 50 the null hypothesis of the asymptotic mean and covariance is
acceptable. Single sets of data were also tested for N equal 100 and
200, and were found to be non-significant.
Values of the test statistic for the unmodified (;CijJ are
given in Table 4.2. On the basis of the combined test values for
t' = 1.0 the asymptotic mean and covariance are acceptable at N = 200
for r = 3, N = 100 for r = 5, and N = 50 for r = 7. For both ~= 0.2
and 2.0, r = 7, the asymptotic parameter values are rejected at N = 50
and accepted at N = 200. However for t' = 2.0 one set of samples
yielded a test value significant at the .01 level and the combined test
value is very near the .05 level. It appears that for this t' the
sample size must be near 200 for the asympt9tic parameter values to
hold, whereas for t'= 0.2 they may hold for some smaller N between 50
and 200.
Table 4.1. Values of the likelihood ratio test statistic of the
hypothesis of asymptotic mean vector and covariance
matrix of convergent (Pij(t')) by r, t', and N
(entries are values from two independent sets of samplesand the sum of the values)
57
t' 0.2 1.0 2.0
~7 3 5 7 7
50 38.69 22.45 20.70 12.90 24.64~
31.81 19.36 32.02 20.25 27.29
70.50 41.81 52.72 33.15 51.93
20 23 .75 54.48** 32.60
9.03 19.32 54.97**...
I
32.78 73.80* 87.57**
10 a 77 .98*** 63.65*** 39.72 40.76
a 131.39*** 29.98 47.12** 46.93**.
209.37*** 93.63*'** 86.84** 87.69**- .,
---
* '** ***Probability < .05, .' P < .01, P < .001
a .Sample dispersion matrix singular, all sample values o~ P23were zero.
Table 4.2. Values of the likelihood ratio test statistic of the
hypothesis of asymptotic mean vector and covariance
matrix of unmodified (~ij) by r, t', and N
(entries are values from two independent sets of samplesand the sum of the values)
58
t' 0.2 1.0 2.0
~7 3 5 7 7'
200 21'.00 22.73 23.34 16.02 19.71
18.29 39.20 30.75 43.80* 50.54**'. "
39.29 61.93 54.09 59.82 70.25
100 50.58** 24.29 15.6~
53.97** 27.58 40.98*>C
f
104.55** 51.87 56.59
50 44.84* 26.30 32.53 42.. 17* 34.46-
46.56*~ 26.29 84.66*** 14.40 79.06***I
, ,
91.40*** 52.59 117.19'A,** 56.57 113.52***
20 45.19* 109.58*** 67.36***-
47.60** 64.95*** 158.66***
92.79*** 174;.53*** 226.02***-
10 98.67*** 73.55*** 145.38*** 59~'66*** 66.85***- - -
192.14*** 46.77* 36.79 61.49*** 74.90***
290.81*** 120.32*** 182.17*** 121.15*** 141.75***.,
*Probability < .05, ** P < .Ol? *** P <.001
59
The values of the test statistic for the modified (tij ) are
given in Table 4.3. With the exception of t' = 0.2 the asymptotic
parameters are not acceptable for even the largest sample size of 200.
For t' = 0.2 they are rejected for N = 50, the next smaller size sampled.
It is apparent that the distribution of the modified estimates does not
approach the asymptotic distribution as rapidly as does ,that of the
unmodified maximum likelihood estimates.
It is not surprising that the asymptotic distribution of the
(tij ) is not an acceptable approximation for sample sizes for which the
asymptotic distribution of the (Pij(t')) is an acceptable approximation.
The t ij are power series functions of the ~ij' the magnitude of the
power varying between samples, depending on the number of terms required
to obtain the desired accuracy of computation. Normality of the joint
distribution of the Pij was only assumed, not tested, in the multivari
ate test. Hence the higher order moments of the Pij may well not have
the pro~erties of the normal distribution. It would seem reasonable
that the higher order moments would approach the normal moments more
slowly than the first and second do and it is conjectured that this is
the factor leading to the rejection of the asymptotic values of the
parameters for larger sample size for the t ij th~n for the Pij.
The null hypothesis of 4.2.1 for the Pij included zero covari
ance of any pair of statistics Plj
' P2k, and was accepted 'for N 2: 50.
It was also of interest to know if the zero covariance of these
statistics held for smaller sample sizes for which the asymptotic
covariance matrix as a whole was not acceptable. The distribution of
Table 4.3. Values of the likelihood ratio test statistic of the
hypothesis of asymptotic mean vector and covariance
matrix of modified (tij } by r, t', and N
(entries are values from two independent sets of samplesan4 the sum of the values)
60
\:
0'.2 1.0 2.0
7 3 .5 7 7I
>
200 26.28 60.j9*** 58.40*** 39.72 84.99***
28.00 60087*** 55.18** 49.70** 140.04***
54.28 121.26*** 113058*** 89042** 225.03***
100 7007.5*** 53.87** 4,3063*
78078*** 57.41,*** 62001***
149 053*** 111. 28*** 105.64***
50 47.20** 92;,42~ 67.22*** 111004*** 123.70***
59.08*** 75080*** 121041*** 57.63*** 149.13***
106028*** 168022*** 188.63*** 168.67*** 272.83***
20 66.60*** 116.53*** 128068***- -
53.24** 151.26*** 173024***-
119.84*** 267.79*** 301.92***- . -
* Probability < .05, **F < .01, *** P < 0001
61
the likelihood ratio criterion 81 to test independence of sets of
normally distributed variables is well-known for the special case of
two sets of 1 and k-1 variables. 281 in this case is equal to l-R ,
where R is the multiple correlation coefficient. Wilks (1935) obtained
expressions for the integral of the lower tail (the rejection region)
of the distribution of 8r for various other special cases, among them
that of two sets of three variables each. He also obtained the
approximate Xl distribution with degrees of freedom equal to the
number of zero covariances included in the hypothesis. The test is
described in the appendix 7.7.2.
This additional multivariate test was applied to those sets of
data for which the asymptotic co~ariance matrix of the (Pij ) was
rejected. Both the exact probabilities and those of the X2 approxima-
tion were computed for several of the smaller sets of data. It was
found that these differed but little and the x2 approximation was
used for the remainder of the data. The results, given in Table 4.4,
indicate that the zero covariance of the (Plj PP2j) is acceptable for
sample size as small as 10.
4.3 Univariate Analysis
4.3.l·Distributions of Effective Sample Sizes
. Evaluation of thetneans, variances and covariance of the ni
• for
the parameter values of the model provided a measure of how well the
MOnte Carlo results represent the true sampling distributions. The
Monte Carlo values from sets of 50 samples are given in Tables 7.6.1 to
7.6.3 with the parameter values. The sample means are very close to
the parameter mean values of the ni •• The sample variances and
Table 4.4. Values· -of the likelihood ratio test statistic of zero
N 10 20
t' r 81pdx?9]
~Pr[ x2
9]
,
0.2 7 a
1.0 3 8.835 > .30
10.299 > .30
5 13 ~561 > .10 13.377 > .10
7 12.188 > .20 12.950 > .10
2.0 7 9.826 > .30
7.617 > .50
asample dispersion matrix singular, a~l.values of ~23were zero.
62
63
covariance differ from the parameter values more than do the means~
This would be expected if the n. are approximately normally distributed1.
since the variance of the sample variance would then be more than double
that of the sample mean.
The marginal probability distributions of nl • and n2• obtained
in section 3 suggested a trend toward normality with increasing sample
size. Hence, the Monte Carlo marginal frequency distributions of the
ni • were compared to normal distributions with the appropriate means
and variances. The two sets of 50 samples for each parameter,point were
pooled. The cumulative marginal distributions of nl • and n2• were
plotted on normal probability paper and compared to the line of the
hypothesized normal cumulative distribution. The Kolmog(jrov-Smirnov
statistic of the maximum absolute deviation between the sample cumula-
tive function and the theoretical could be graphically determined to an
accuracy of .01. This was sufficient since the sample function was
based on only 100 samples. The value of the maximum deviations are
given in Table 4.5 with si.gnificance levels indicated by asterisks.
For the nL only one maximum deviation is of significant magni
tude and that is for sample size 200 for r = 3. This result is presumed
to be a Type I error and the hypothesis of normal marginal probability
distribution of the nl • is accepted for all sample sizes of 5 or larger.
Normality of the marginal distribution of n2• is rejected for sample
size 5 for r = 5, and 7, and for N ~ 10 for r = 3 since all test values
are significant at the .01 level or beyoI).d. For r = 7, N = 20 the
test value is just significant 'at the .05 level, and for N= 10 the
test value does not approach significance. Hence the nClrmality of n2. '
is accepted for N ~ 10 for r = 5, 7, and for N ~ 20 for r = 3.
64
Table 4.5. Values of the Kolmogorov-Smirnov test statistic of
the hypothesis of normal marginal distributions of
the effective sample sizes with means and variances
determined from probability generating function
evaluation for t' equal 1.0, by sample size and by r
.~ 3 5 7
nl. n2• nL n2 n1 n2
5 .13 .18** .11 <. .17** .06 .22**
10 .05 .25** .07 .11 .10 .07
20 .06 .09 .09 .07 .08 .14*-
50 .10 .12 .04 .09 .07 .07
100 .08 .08 .08 .10 .09 .10
200 .15* .08 .10 .06 .08 .07
*Probability < .05, **Probability < .01
65
4.3.2 Range and Frequency of Negative (tijJ
The range and frequency of negative t .. are given in Table 4.61.J
with the parameter values A.ij • In general, with increasing N there is
a decrease in both the range of negative values and the proportion of
convergent samples yielding them. Irregularities in the pattern for
the smallest sample sizes are understandable.in light of the conditions
which give rise to negative estimates. A Pij equal to zero may yield
a negative ~ .. only if Pl' and PZ' are not both zero. For the smallest1.J J J
sample sizes both P13-and P23»- or P14 and PZ4' were zero in a larger
proportion of the convergent samples, and hence a lesser proportion of
the samples yielded negative estimateso The N at which a regular trend
of decrease in range and frequency of negative values begins varies with
the ~ij and with r.
Both the range and frequency of negative values of the t lj are
markedly smaller than those of the t 2j o This is another consequence of
the greater amount of information on transitions from Sl resulting from
the initial condition of all members of the sample in SI. The maximum
deviations of the marginal distribution!,!. of the P2j from-the asymptotic
normal, which were greater than tho~e of the Plj
' almost always o~curred
at zero and were positive, that is, the proportion of zeros exceeded the
asymptotic probability. The marginal distributions of the Plj , on the
other hand, showed an excess of zeros in only one case.
Few negative values occur for t13
with parameter value 0.18,
whereas for t l4 with parameter value 0.05 negative values occur for
sample sizes up to 50. Similarly the range and frequency of negative
values of ~24 with parameter value 0.10 are much smaller than those of
Table 4.6. Range and frequency of negative values of (~ij)' by t',
rand N
66
t13 t 14 t 23 t 24
,Parameter value .18 .05 0 .10
t' r N W a neg. neg. neg. neg.cn range fre,q. -range freq. range freq. range freq.
0.2 7 10 77 -.03 5 -.09 25 -.03 850 100 -.01 1 -.04 69 -.02 8
200 100 ·'.02 70
1.0 3 5 44 =.05 1 - .37. 18 -.18 1010 59 -.05 2 -.11 2 -~39 27 -.26 1720 89 -.05 9 -.30 55 -.08 3850 98 -.03 2 -.15 72 -;06 28
100 100 -.11 65 -.03 12200 100 -.09 49 -.02 2
5 5 60 .... 05 1 -.09 10 - .31 38 -.15 810 81 -.19 8 -.34 57 -.19 2820 94 -.06 7 -.30 62 -.06 3050 100 -.11 57 -.04 9
100 100 -.10 47 -.01 1200 100 -.07 42
7 5 55 -.12 9 - .32 36 -.09 1310 92 -.03 8 -.22 69 -.15 3620 99 -.04 2 -.19 65 -.09 2850 100 -.11 55 -.01 2
100 100 -.07 54200 100 -.04 48
2.0 7 10 61 -.07 1 -.13 9 -.36 33 -.15 2450 100 -.01 3 -.27 53 -.03 1
200 100 -.12 45
aNumber of convergent samples out of 100.
..
'..
67
t 23 with parameter value ~ero. The frequency of negative values of t 24
decreases to zero for the larger sample sizes except for t' = 1.0, r = 3.
Negative values of t23
, although of small absol~te magnitude, occur in
substantial numbers for even the largest sample size of 200. The
probability of a negative value of a particular t ij appears to depend
both on the magnitude of the parameter and on whether or not the
instantaneous rate is that of a transition out of the initial state.
4.3.3 Ko1mogorov-S~irnovTest of Marginal Distributions of {Pij } and
The graphic Kolmogorov-Smirnov method was also used to test the
conditional marginal frequ;ency distributions of the Pij and t ij from
convergent samples against normal distributions with the means and
variances·specified by the as~ptotic distributions. The frequency
dis tr1but ions were computed from the combined convergent samples of the
two sets for each parameter point and thus were based on 100 or less
1s 0.50. Hence it is obvious that for any finite N the distribution of
the modified t23
cannot approach the as~ptotic d1strib~tion. The
68
empirical distribution function of a set of sample values of modified
r23 will always deviate from the asymptotic normal by an amount greater
than or equal to 0.50. Thus the Kolmogorov-Smirnov test will always
lead to rejection of the hypothesis at the .05 level if the set con-
tain~ seven or more sample values, and at the .01 level if it contains
ten or more values.
Graphs of the marginal cumulative distributions of ~23 by
sample size for t' ,... 1.0 and r = 7 are given in Figures 4.1 to 4.3 as
examples. The maximum absolute deviations for the Pij are given in
Table 4.7 and for the modified and unmodified t ij in Tables 4.8 and
4.9, respectively. The number of convergent samples on which ,the
empirical functions were based ar~ included. together with.the critical
values of the Kolmogorov-Smirnov statistic at the .05 level from
Birnbaum's tables (1952) of the exact distribution for finite sample'
size. The significant deviations are blocked to indicate the pattern.
of results.
'It is apparent that the asymptotic marginal distributions of
thePlj hold for sma~ler N than do those of the P2j' There is ~9me
progression with increasing r, the asymptot~c distributions of the Plj
being acceptable approximations fer r = 3 for N ~ 20 and for r= 5, 7
for N ~ 10. The asymptotic marginal distribution o£~2l is also an
acceptable ap~roximation for N ~ 10 forr = 5, 7, but only for N ~ 50
for r = 3. For P24 the asymptotic distribution is an acceptable
appro~imation forN ~ 50 for r = 5, 7, and for N ~ 100 for r = 3. The
marginal distribution of P23 tends to the asymptotic most slowly, being
acceptably approxim~ted by it only for sample size 200 for all r
sampled.
e •• e. .
e
asymptotic normal .70
x-unmodifiedestimate
N=IO
asymptotic normal
maximumdeviationmodifiedestimate
.10
.05
x- unmodifiedestimate
.60
.50
40
.30
.90
.80
.95
.20
Probability.99., (normal scale)
maximumdeviationmodifiedestimate
N=5
xx
.1maximumdeviation I
unmodifiedestimate
~
.10
.05
.95
Probability.99., (normal scale)
.70
.90
.30
.20
.80
.60
.50
40
.01' 'I I i 'i i I I I .01" i I I i I I I-40 -.30 -.20 -.10 0 .10 .20 .30 40 -.20 -.10 0 .10 .20 .30 40
A AX23 X23 (J\
\0
Table 4.1. Asymptotic normal and Monte Carlo marginal cumulativeA
distributions of ~23' t' equal 1.0, r equal 7, N equal 5,10
e.•
, . e..
e
maximumdeviationmodifiedestimate
-, asymptotic normal
o .04 .08 .12/I
A23
maximumdeviation
unmodified testimate I
.10
.70
.x-unmodifiedestimate
.05
.60
.50
40
.90
.95
.30
.80
Probability N = 50.99, (normal scale)
.20
asymptotic normal
mallimumdevIationmodifiedestimate
N=20
.01 v I I " I io .04 .08 .12 .16 .20 .24 -.08 -.04/I
A23
.01 f iii 'i iii i I I-.16 -.12 -.08 -.04
.95
.90
.80
.70Jmaximumdeviationunmodified
.60 Iestimate
.50
«>
.30
.20
.10
.05
Probability.99, (normal scale)
Table 4.2. Asymptotic normal and Monte Carlo marginal cumulative/I
distributions of ~23, t' equal 1.0, r equal 7, N equal 20,50 .....o
e•. . .
e e
.01' i i 'I I Iii
-~~4~ 0 m M ~ 00A~23
.80
asymptotic normal
maximum
Ideviationmodified
FII
N=200
x
maximumdeviation
unmodifiedestimate
x-unmodifiedestimate
.01 Y I 1 i '. iii-.08 - 06 -.04 - 02 0 .02 .04 .06
A
~23
.95 .,
.10
.05
20
.70
.80
.60
.90
.50
40
Probability99., (normal scale)
.30
N= 100
maximum
I deviationmodified
. estimate
rIII
maximumdeviation
unmodified Iestimate..,
x-unmodifiedestimate
70
.10 -I x
.95
.05
.30
.20
.90
.60
.50
40
Probability.99., (normal scale)
Table 4.3. Asymptotic normal and Monte Carlo marginal cumulative"distributions of ~23' t' equal 1.0, r equal 7, N equal 100, 200
.....I-'
Table 4.7. Values of the Ko1mogorov-Smirnov test statistic of the
72
hypothesis of normal marginal distributions of convergent
(Pij(1.0)) with asymptotic means and variances, by sample
size and by r
r
3
N
5
10
20
50
100
200
W acn
44
59
89
98
100
100
.20
.17
.14
.13
.13
.13
r - - .-...- --" - -...& ~ -- ""- --- ---..
1
·36 . .27 .39 .28 .55.60 11·
r----lL2~ .09 .17 1.31 .53 .51 I.08 .11 .13 L.:1!.... l .43 .40 I.05 .07 .05 .10 r .47 .21 Ir _. --'.05 .05 .09 .10 L.:3~ .10
.05 .15 .05 .10 .12 .09
.08
5
7
5
10
20
50
100
200
5
10
20
50
100
200
60
81
94
100
100
100
55
92
99
100
100
100
.17
.15
.14
.13
.13
.13
.18
.14
.13
.13
.13
.13
r::lL:.2~
.15
.08
.08
.09
.09
r-'::JL..2~
.09
.08
.06
.08
.07
.10
.09
.05
.05
.08
.09
.10
.10
.06
.07
.07
.07
" ,......,1 ·43 .22 .56 .47 'I-' - - -,.13 .13 r .52 .40 I.14 .14 ,.40 r-=-2~
.07 .05 .29 .06
r :l I IL..:2~ .09 L:.l~ .07
.04 .06 .04
" - - -- ~r .21 .20 .50 .33 I1...----
1.08 .07 1'.45 .29
1.01 .07 I .41 r --.:1?.-J
•07 .05 I' 24 I .05
•09 .08 L: 17-l .05
.05 .10 .07 .08
aNumber of convergent samples out of 100.
bKolmogorov-Smirnov critical value at .05 level.
73
Table 4.8. Values of the Kolmogorov-Smirnov test statistic of the
hypothesis of normal marginal distributions of the modified
(tij } with asymptotic means and variances, by sample size
and by r,. for t' equal 1.0.
a b t 12 t 13 t 14 t Z1 t 23 t 24r N tv d .05cn
r--.- -- -- --'- -.-3 .5 44 .20 1..:..27 .Z2 .34 .28 .50 .57 ~
10 59----I I
.17 .14 .07 ~20 .25 .50 .47-- ---I I
20 89 .14 .10 .10 .11 .13 ,.50 .36 I50 98 .13 .11 .07 .12 .09 ,.50 .l~
100 100 .13 .07 .07 .04 .12r-'
.50 .13
200 .13 .06 .04I :-1
.07I L_...,
100 L.:.l~ 2 0__.~':J
5 .5 60 .11r:l .14
r-- -- ~--,
~2~ 1.38 .20 .50 .35----l I
10 81 .15 .13 .09 .15 .13 .50 .38
94 .14 .14 .04,- :I I I
20 I.21 I .06 I.50 r -.:...21...J
50 100 .13 .10 .09 •20 .07 •50 •11I L __ .J I100 100 .13 .09 .07 .16· .15 .50 I .05
.04I ,--I
.08200 100 .13 .05 1-2-1~ .05· ~5Q.j
.18,', r ---I
7 .5 55 .17 .10 j .21 .17 1.50 .31,r- I
10 92 .14 .13 L:1~_.1~ .09 ,.50 .26,
20 99 .13 .12 .08 .13 .13 1.50 .18 Ir--50 100 .13 .05 .11 .12 .07 .50 .07
100 100 .13 .13 .07 1 271 .12I ·1
.06.50
200I. I I I
100 .• 13 .08 .09 L.~~I .10 L:5~ .12
aNumber of convergent samples out of 100.
bKolmogorov-Smirnov critical value at .05 level.
Table 4.9. Values of the Ko1mogorov-Smirnov test statistic of the
hypothesis of normal marginal distributions of the
unmodified (rij ) with asymptotic means and variances, by
sample size and by r, for t' equal 1.0
a br N W d. 05 r12 r13, . r14 t 21 t 23
r24cn
3 5 44 •20r'--- -:r r ---- -,L:22' .34 . .•47 .57 ,
-_.--J I I10 59 .17 .07 .11 .40 .47 I1. 2820 89 .14 .10 .36
1I50 98 .13 .12 .27 .18
I r-. ....J
100 100 .13 .25 1 .13L_,
r":J200 100 .13 .11 L':1~
5 5 60 .17 .14r~ r:-- -I
L3~ 1.45 .35,
10 81 .15 .09 1.38 .38 I94
,- -:-l20 .14 J...:2~ .21 .16 I
t r--50 1QO .13 L:-1~.J .11
100 100 .13 .11 .05
200 100 .13 .08r-, r--:-1
7 (5 55 .18 t-.: 1!J 1.40 .29 I':io 92 .14 .07 .29 .26 I
I r~-20 99 .13 .13 1. 20 r .12
50 100 .13 1. 151
.05
100 100 .13 .19I I
200 100,) .13 L:.1~
aNumber of convergent samples out of 100.
.bKolmogorov-Smirnov critical value at .05 level.
74
75
The pattern of results for the unmodified and modified ~ij
differs only for t23
• The asymptotic normal distribution is an
acceptable approximation for the unmodified estimate for N = 200, r = 3,
and for N ~ 100, r = 5. As indicated previously, the normal distribu
tion does not hold for the modified t 23 for any sample size. For both
modified and unmodified t l2, Al3 , and t 2l the asymptotic normal distri
butions are generally acceptable approximations for N ~ 20, r = 3, for
N ~ 10, r = 5, and for N ~ 5, r = 7. The several exceptions are
attributed to Type I e~ror since they occur for larger sample sizes.
In considering the large deviations of the marginal distribu
tions of A14 it dshould be noted that the standard deviation of the
asymptotic distribution for the larger sample sizes is of the order of
magnitude of the computational accuracy of the AijJ .01. The grouping
of the sample values imposed by the computational limit is too coarse
to provide a valid test of the approximation by the asymptotic distri-
bution for the larger sample sizes.
It may be recalled .that the purpose of sa~pling for t' equal
0.2 and 2.0 was to investigate the possible effect of the magnitude of
the Pij(t') on the sampling distribution of the estimate. The
Kolmogorov-Smirnov test values of the distributions of the pij(t') for
the parameter points which were sampled for all three time periods are
given in Table 4.10. For any ij, the value of Pij(t') increases with
t' and the results of the tests show that the sample size for which the
asymptotic milrginal distribution of that Pij is acceptable decreases.
For fixed t' a similar relationship may be observed for the P2j' in
that
Table 4.10. Values of the Ko1mogorov-Smirnov test statistic of the
hypothesis of normal marginal distributions of conver
gent (Pij(t ' )} with ~symptotic means and variances, by
sample size and by t ' , for r equal 7
,
t I N Wa d
0 A A A "·A A A
en .05 P12 P13 P14 P21 P23 P24
1- --, ,..--- - - -',0.2 10 77 .15 r.:1~ .14 1'.47 .43 .54 .52 ir- - -,
50 100 .13 .06 .09 L~~' .08 1.54 r _.3~
200 ·100 .13 .05 .08 .07 .06 L:4!J .08
.... r --:I1..0 10 92 .14 .09' .10 .08 .07 .45 .2~Ir- .
50 100 .13 .06 .07 .07 .05 L.2~ .05
200 100 .13 .07 .07 .05 .10 .07 .08
61 .17 .14 .09 .06,- -------.
2.0 10 ~18 .32 .2!J----50 100 .13 .08 .05 .06 .11 .07 .08
200 100 .13 .10 .10 .04 .06 .05 .07
aNumber of convergent samples out of 100.
~Kolmogorov-Smirnov critical value at .05 level.
76
77
and the sample sizes at which the asymptotic distributions of the
estimates are acceptable are invers·ely ordered.
The Kolmogorov~$mirnov test values for the ti,j for the three
time periods are given in Tables 4.11 and 4.12 for the modified and
unmodified estimates. .The asymptotic distributions of t 13 , t 2l , and
t 24 are acceptable approximations for smaller sample sizes with
increasing t~ Again the difference ~etween the results for the modi
fied and uD&odified estimates is limited to t23
• The asymptotic
distribution of the unmodified ~23 is an acceptable approximation for
N ~ lOfor t' = 2.0.
4.3.4. :Bias
Two estimates of the bias of Pij and r ij were obtained for each
parameter point through the &enerationof the samples in two separate
sets of SO each. The difference between each pair of estimates pro-
vided at least a minimal estimate of their variance and it appeared
-1roughly proportional to N • The data on bias are given in Tables
7.6.4 to 7.6.8.
Two methods of judging the significance of the bias were con-
sidered, one based on the individual sets of samples and the .other on
-1the regression o~ the estimates on N over the range 5 S. II S. .200.
Considering a single set of ~amples, let Y denote anyone of the
statistics and Yk a sample value where
k = l;2, ..•W; WS. 50,
and the samples are independent. If the marginal distribution of the
statistic is even approximately normal, the mean of r,oughly 50 samples
will be verynear~y normally distributed. Assume that Y has a mean of
Table 4.11. Values of the Ko1mogorov-Smirnov test statistic of the
hypothesis of normal marginal distributions of modified
(tij )w[t'h asymptotic means and variances" by sample
"size and by t'" ~or r equal 7
a bt I N W d .05 t l2 tl:3 t 14 t 21 t 23 t 24cn
,- --- -- --" --'::-10.2 10 77 .15 .13 L:l~ .46 043 l .50 .52 r
l r- -50 100 013 005 011 ].14, .08 1.50 r"_·3~
200 100 .13 .12 011 L.:1~ .06 L..:.5~ .07
1.0 92,- ---,
009,- --,
10 .14 013 ~16__.1~ r .50 1--:3~50 100 .13 005 .11 012 007 050 .07
r TI I I200 100 013 008 .09 L:2~ .10 L5~ .12
10 61 .16r -- . ....,
2.0 017 012 007 .16 050 02~
I-l~I ,--'
50 100 013 .05 .10 006 1.50 I .08
200 100 .13 010 007 L.2~ .04 L:5~ .08
aNumber of convergent samples out of 1000
bKolmogorov-Smirnov critical value atoOS level.
78
Table 4.1Z. Values of the Kolmogorov-Smirnov test statistic of the
hypothesis of normal marginal distributions of
unmodified (tij } with asymptotic means and variances,
by sample size and by t ' , for r equal 7
a bt I N W d .05 A1Z "'13 Al4 "'21 AZ3 AZ4cn
,', r - -'-'0.2 10 77 .15 1.46 I .. 50 .52 .
I, I50 100 .13 .14 .47 .3~
I I I 1-zoo 100 .13 ~~~ ~3~ .07
1.0 10 92 .14r--.··.I
.07 .29 . Z6-1
50 100I r-"
.13 .12 .15 .. 05
ZOO 100 .13,- ...., I I
.1Z'L:2~ L:l~
Z.O 10 61 ~17 .05 .1Zr·...,.2~1-.
50 100 .13r,
.10 .081.18 IZOO 100 .13 .08 .08L:2~
a .Number.of convergent samples out of 100.
bKolmogorov-Smirnov critical value at .05 level.
79
80
2the form ~~ and variance ~ , so that
2y -+ [~~, ~ . ]
The usual confidence limits on the mean of Y can be written
Pr[yst (W-l) _ . sta(W,;,l)
]ex 1 (4.1)<~+f3<Y+ . = - exrw rw "
2where s is the sample set estimate of the variance and tex(W-l) the
appropriate ex significance point of the t distribution with W-l degrees
of freedom. If ~is a given non-negative value, subtracting it through-
sta(W-l), =< (Y - ~) + IT; :I 1 - a
out leaves the inequality unchanged
st (W-1)Pr[(y -~) - a,." < f3
li
so that
(4.2)
The estimates obtained of the bias of the statistics Pij(t ' ) and ~ij
were of the form (y -_~) and 95~ confidence limits were computed in the
program, an interval not coveri~g zero being equivalent to 'rejection of
zero bias at the .05 level. It was found that,no consistent fattern of
the bias with respect to sample size emerged from these results.
An alternative method of judging the bias from the regression.. -1
of the estimates on N was then carried out. For a given r andt' let
Yi denote anyone of the statistics which is obtained from a sample of
size Ni , and let ~ be the parameter value of which Yi is a consistent
estimate, so that the mean of Y., as N., becomes infinite, is~. For]" ],
N. finite, assume that], ,
E(Yi
) = ~. + ~ ,I Ni2
~ -and that the variance of Yi is~. Let Yik denote the mean of a set ofi
Wik values of Yi • The bias estimates were then
-bik := Yik - ~, k =1,2;i := 1,2, ... ,6 •
81
As was stated above, Yik and hence, bik may be assumed nearly normally
distributed so that2
bik -+ n[ tLN ' w(J' N ]i ik i
(4.3 )6
I...~ ...
that value ~ which gives6 2
min \ \~ ~ ~ Ni(bik
i=l k=l
6 2
I I b iki=1 k=l
Inclusion of Wik, the number of samples in the set, in the denominator
of the variance of bik would have increased the comp~tational labor in
the application out of proportion to the amount by which the Wik
varied from 50. . Hence, tke model used was
tL (J'2 ]bik -+ Y
82
and the normality of bik, Q1 and Q2 are independently distributed as
chi-square variables with 11 and 1 degrees of freedom, respectively.
Under the hypothesis ~ = 0, the ratio
will have an F distribution with land 11 degrees of freedom and pro-
vides a test of the hypothesis.
Although the asymptotic marginal distributions were acceptable
approximations for the larger sample sizes, it was judged better to fit-1 ,'-.
the regression of bias on N to the data on all sample sizes. The
estimates of the regression coefficients ~ are given in Table 4.13 for
the Pij and in Table 4.14 for thetij •
Regression coefficients of bias significantly different from
zero obtaine4 with consistency over the sampled parameter points (t~r)
only in the cases OfPl2, t 12 and t 23 • In these cases the bias of the
statistic is posit~vefor all (t', r) and the regression coefficients are
ordered over both ~ and r, with the greatest bias for smallest r for
t' = 1.0, an,d for lar$est t ',: for r = 7. In the case of other.
statistics for which s:i.gnificant coefficients obtained for only one
or two parameter points, consistency is lacking in the sign and order
of magnitude of the coefficients. It is concluded that bias has not
been demol'istratedfor these statistics., '.
Negative values of t l2 do not occur. Hence the bias of that
statistic can not be attributed to the modification. In the case of
t 23 the regression coefficients for the unmodified estimates were
negative and not significantly different from zero. Thus the
Table 4.13. Estimates of coefficients of bias regression on
reciprocal of sample size, and significance of
difference from zero, for convergent (Pij(t'}),
by t' and r
~ 0.2 1.0 2.0
3 .244**A 5 .161**P12
7 .041* .158** .158"
3 -.082 ; ,
A 5 .011P13
7 .014 -.021 .121
3 ,.049,.,
5 -.021P14
7 -.014 .010 .042
3 -.360A 5 -.209*P21
7 .284 -.113 .592**- -
3 -.104
A 5 -,,051P237 -.010 -.030 .091
3 -.268**A
...
P24 5 .111
7 .147 .009 .257
* ** ***Probability < .05, P < .01, . P < .001
83
Table 4.14. Estimates of coefficients of bias regression on
reciprocal of sample size, and signif~cance of
difference from zero, for modified {~ij}' by
t I and r
~ 0.2 1.0 2.0
3 0.62**
t 12 5 0.52**
7 0.29* 0.40** 0.56**
3 0.03
t 13 5 0.12
7 0.03 0.04 0.21
3 0.10*
t 14 5 -0.00
7 -.0.Q8 0.03 0.04
3 0.52
AZl 5 0.46
7 1.42 0.44 0.11
3 0.42*
t 23 5 0.34*
7 0.14 0.35** 0.82**- -
3 -0.20
t 24 5 0.36*
7 0.52 0.16 0.46
* ** ***Probability < .05, P < .01, P < .001
84
""
85
modificationdges introduce a small but significant positive bias of
t 23 as would be expected for an estimate with parameter value zero.
4.3.5 Variance
The sampling plan also yielded pairs of estimates of the
variance and coyariance of Pij and t ij • The data are given in Tables
7.6.9 to 7.6.34. The significance of the data was assessed by two
methods analogous to those for the bias data. The equality of the
small sample variances to the asymptotic values was tested for each
set of samples by reference to the chi-square distribution with degrees
of freedom dependent on the number of convergent samples in the set.
The variance estimates which differed significantly from the asymptotic
values again followed no d;scernible pattern with respect to sample
size.
Regression techniques were then employed to fit and test the
hypotheses that the small sample variances of the statistics are of the
form
..:L +-LN N2
where VN-l is the variance specified in the asymptotic distribution.
2LetSi denote the usual sample variance estimate" wh.ere each sample
2value has a size Ni " and sik the value of the estimate obtained from
a set of Wik sample values... Let
2 Vsik - Nt •
It was assumed.that vik is approximately normally distributed with
mean 1N..2 and variance cr~-2. The sample variance estimate sik is
distributed as a chi-square variable with k-l degrees of fre~dom
multiplied by the true variance. The chi-square distribution
86
approaches the normal rather slowly so that for k equal 50 or less the
approximation would not be especially good. However, the 'hypothesis
tested in the regression model is that of a mean, i.e., the mean of
variance estimates and it is known that tests of hypotheses for means
are robust. Thus, the model was considered reasonable as well as
convenient.
The estimate of ., and the test statistic for the null hypothesis
A
r
of zero are, by derivation analogous to that of section 4.3.4,
6 2
I I V ik,- 1::1 k=l
"'" 6
I,;2i"-=1 i
and
(4.5)
Iz
....2 -111 N~
• 1.1. (4.6)
The., estimates for the (Plj(t')) and (tij ) are given in Tables
4.15 and 4.16 with sig~ificant difference from zero indicated by
asterisks. In the case of Pl" the coefficient of N-2 is significantlyJ -
different from zero for only one parameter point, and that coefficient
is very small. It is concluded that the asymptotic functions of N- l are
acceptable approximations to the variance of the (Plj
) for finite sample
size. For each of the (PZj ) the regression coefficients of N-2 differ
from zero significantly for two or three of the five parameter points.
Recalling (Tables 7.4.2 to 7.4.6) that the asymptotic variances of the
(Pij ) decrease as r increases for t' equal 1.0, and decrease as e
Tabl~ 4.15.
87
. -2Estimates of coefficients of variance regression on ~ ,
and significance of difference from zero, for convergent
(Pij(t')),by t' and r
~!
0.2 1.0 2.0
3 -0.032A 5 0.013P12
7 -0.020* -0.011 0.093
3 0.027A 5 0.008P13
7 -0.001 -0.024 -0.123
3 0.014A 5 0.020P14
7 0.004 0.017 -0.058
3 -2a88A
5 -1.583***P21
7 2.259*** -0.567* -6.458..
3 "'0.488**A 5 -0.169**P23
7 -0.033 -0 ..034 0.112
3 -1.593***A-
5 -0.026P24
7 1.466* -0.186 0.781
* ** ***Probability < .05, P < .01, P < .001.
Table 4.16.
88
-2Estimates of coefficients of variance regt;ession on N ,
and significance of difference from zero, for modified
~ 0.2 1.0 2.0
3 0.95
~12 5 0.84**
7 -0.57* 0.47** 0.73*,
3 0.36**"
~13 5 0.16**
7 -0.29 0.19 0.07
3 0.10**
~14 .5 0.10
7 0.10 0.12* -0.00
3 -4.20
:t21 5 0.63
7 33.26 0.56 -5.62
3 -1.12**
t23 5 -0.48*
7 -1.08** -0.16* -1.08
3 -2.80**
t 24 5 0.51
7 21.34** . -0.07 0.47,"
*Probability< .05, **p < .01, ***P < .001
89
decreases for r equal 7, the order of the regression coefficients
appears reasonable for P21' and with the exception of the parameter
point (2.0,7) for P23 and P24' Hence, it is concluded that the
-2addition of a term in N would significantly improve the asymptotic
approximations for the variances of the (P2j)'
Similar to the results for bias, a fairly consistent pattern,of
-2 ~significant coefficients of N obtained for the variance of A12 and
t23
, so that the addition of the term would improve the asymptotic
approximatio,ns. Since none 'of the regression coefficients i~ signifi
cant for t 2l, the asymptotic approximation to the variance is considered
acceptable for finite sample size. Significance of the coefficients for
two of the five parameter points for each of the remaining tiJgives
some evidence that an additional term would improve the approximations
to the variances. However, in the absence of a regular order of magni-
tude of the coefficients, the resuits are considered inconclusive.
The regression coefficients for the variance of unmodified ~23
were non-significant, so that the asymptotic variance 'is accep~able for
the unmodified estimate. As would be expected, the introduction of
bias of t23
by the modification is a~companied by a significant'reduc
tion in the variance of the estimate.
4.3.6 Covariance
The same model of regression analysis was used to fit and test
-2'sigqificance of an additional term in N for the covariances, ~f the
(Pij) and of the (tij
). Normality assumptions are perhaps more ques
tionable in the case of covariance estimates but the oft employed
justification that no more reasonable approach could be found applies.
90
The regression coefficients with significance levels indicated are given
in Tables 4.17 to 4.19.
For the (Pij} the few instances of significant coefficients are
discounted in view ~f the generally small magnitude and the absence of
ordering. It,is concluded that there is no evidence that an additional
-2term inN would improve the-approximations to the, covariance of t:he
(Pij,Pik) for finite sample size.
-2The number of instances of significant coefficients of N for
the cov.ariances of the (fijJ indicates that the asymptotic approxima
tions are not_generally acceptable for finite sample size. The occur-
rence of the significant coefficients is too irregular to permit(
conclusions as to which covariance approxima~ionswould be improved by-2 .
the addition of a term in N •
4.3.7 Mean Square Errors
The mean square errors and mean product err?rs~ of the (Pij)
and modified (~ . .) are included, Tables 7.6.35 to 7.6.47, for1J
complet~ne~s. Apart from considerations ~f normality, the mean square
errors provide the best practical basis of comparison of the modified
and unmodified (tij ). From Table 4.20 it can be seen that the mean
square ,error of the modified t ij was less than that of the unmodified
estimate in every case, as was to be expected theoretically.
Table 4.17.
91
-2Estimates of coefficients of covariance regression on N ,
and significance of difference from zero, for convergent
.~ 0.2 1.0 2.0
3 -0.016.... .... 5 0.052**P12,P13
7 0.001 0.007 -0.113
3 0.003.... .... 5 0.002P12,P14
7 0.004* 0.003 -0.002
3 0.012A A
5 0.036P13,P14
7 0.001 0.010 0.045
3 0.091A A 5 0.055PZ1,P23
7 0.005 -0.010 -0.459
3 0.524*A. A 5 -0.004P21 ,P24 .,
~
7 -0.340* 0.060 -1.445**
3 0.031A A. 5 0.002P23,P24
7 0.001 0.018 0.057
*Probability < .05, **p < .01, *** P < .001
Table 4.18.
92
-2Estimates of coefficients of covariance regression on N ,
and significance of differen~e from ,zero, for modified
~.'
0.2 1.0 2.0
3 0.32*:
t12,:t13
5 0.31***
7 0.08 0.11 -0.71
3 0.28
A12,:t14 5 0.15
7 0.00, 0.20 0.08
3 0.10*
:t13 ,A14 5 0.01
7 -0.09 0.03 0.13
3 1.92~
A2i~A23 5 1.03**
7 1.40 0.72 2.17
3 0.22',
A21'A24 5 0.70
7 -7.17 0.20 -2.02
3 -0.07
A23 ,A24 5 0.04
7 -0.05* 0.06 0.10
* ** ***Probability < .05, P < .01, P < .001
Table 4.19.
93
-2Estimates of coefficients of covariance regression on N ,
and significance of difference from zero, for modified
~ 0.2 1.0 2.0
3 1.76
~12'~21 5 1.62**
7 1.56* 0.54 -0.90
3 0.08
~12,t23 5 0.11
7 0.00 0.07 0.30
3 -0.08
~12'~24 5 0.09
7 -0.20 0.08 ) 0.21
3 0.60
~13'~21 5 0.06
7 0.38 0.01 -0.23
3 0.17*
t13'~23 5 0.12*
7 0.10 0.12** 0.43
3 -0.01
~13'~24 5 0.15
7 1.40 -0.07* -0.16
3 0.45*
~14,t21 5 0.087 1.49 0.02 0.45
3 -0.01
~14,t23 5 -0.057 0.02 -0.04 -0.08
3 0.24*
~14'~24 5 0.017 0.49 0.04 -0.44
*Probability < .05, **p < .01, ***P < .001
94
Table 4.20. Estimates of mean square errors of modified and unmodified
{~ij) from convergent samples in sets of 50 for r equal
7,_ by t' and by sample size
t' N t 14t
23t
24
unmod. mod. unmod. mod. unmod. mod.
0.2 10 .004 477 .004 273 .000 570 0 .349 962 .349 327
.007 761 .007 638 .000 669 0 .323 147 .322 811,
50 .001 010 .000 988 .000 695 .000 522 .029 170 .028 700
.003 929 .001 291 .037 094 .036 879
200 .000631 .~ 543
.000 706 .000 622,.
1.0 10 .002 068 .001 913 .023 463 .020 120 .035 189 .032 373"
.002 006 .001 871 .005 786 .001 784 .027 017 .024 913,
50 .002 710 .001 536
.001 873 .000 816
200 .000 482 .000 249-.
•000 388 .000 256
2.0 10 .004- 895 .002 109 .025 058 .012 026 .049 276 .042 746
.003 111 .002 701 .023 537 .017: 413 .017 ;424 .013 951
50 .000 640 .000 618 .007 167 .003 646 .006 134 .005 989
.000 675 .000 577 .005 922 .003 554 .008 214 .008 214
200 .001 171 .000 358
.001 253 .000 722
5. DISCUSSION AND CONCLUSIONS
5.1 Discussion
5.1.1 Generality of Results of the Study
5.1.1.1 Model. The four state model, two transient and two
absorbing, of the Monte Carlo study was the simplest which would
suffice for a follow-up study. Intuitively, with an increase in the
numbe~ of states in the model, and hence in the number of parameters
to be estimated, the sample size necessary for a given precision would
increase.
Addition of absorbing states to the model would only increase
the number of components in the two sets of multinomial-like p..l.J
estimates and decrease the magnitude of their mean values somewhat.
Thus the effect on the properties of the Pij and the t ij would be
minor, and the conclusions from this Monte Carlo study may be general-
ized to models with two transient states and two or more absorbing
states.
Addition of a transient state to the model, on the other hand,
would introduce another set of multinomial-like estimates and another
way in which an incomplete sample could occur. The total information
from a complete sample would be divided into three rather than two .
parts, namely, the effective sample sizes for the three sets of~
estimates Pijo The investigation of the model with two transient
states showed the division of information from a sample to be unequal,
with n2• small relative to nl. ,i for a three trans.ient state model n3 •
96
would also be small relative to nL ." The properties of the P3j would
be expected to resemble those of the P2j of the model studied rather
than those of Plj • The additional division of the sample information
would be expected to increase the sample size necessary for comparable
properties of the estimates in a model with three or more transient
states.
An expression for the probability of an incomplete sample could
be obtained for a model with more than two transient states. It would
be a function of the probabilities of transitions from all the transient
states rather than from only the initial state as is the case for the
two transient state mode1~ Consequently prior information on more
transition probabilities would be needed to estimate the probability
of an incomplete sample for given sample size.
As indicated previously, if the sufficient condition for con-
vergence of the t series estimates
Pii(t') > 1/2, for all i,
did not hold, in a model with three or more transient states the
necessary condition would require evaluation of the coefficients of the
characteristic equation of ~*(t'). The bound on the remainder term of
the series was based on the pattern of convergence peculiar to the form
of ~*(t') for two transient states, and hence the validity was not
established for models with more than two transient states.
5.1.1.2 Initial condition. The initial condition in the study
was that all members of the sample were in the same state. Two effects
of this were the possibility of an incomplete sample and the unequal
division of the information from a complete sample. The major portion
97
of the information contributed to the Pl j estimates and a very small
portion contributed to the PZ' estimates. This resulted in the. J
properties of the P1j being much better than those of the PZj for
given sample size. The same result was observed for the Alj and A2j ,
although to a lesser degree since they are functions of both the Pl j
and P2j' If the members of the sample could initially be divided
among all the transient states, an incomplete sample could not occur
and the sample information would be more equally divided among the
sets of estimates Pij' 8i transient. The properties of these sets of
estimates and of the Aij would be more similar for all i. It is
possible that a more balanced division of the sample information would
result in an overall improvement in the properties of the A:ij for
given sample size, but this conjecture would require further investi-
~tion. The conclusions from this Monte Carlo study apply only to
samples with all members in the same initial state.
5.1.1.3 A Parameter Matrix. The A parameter matrix investi-
gated in the Monte Carlo study was chosen to yield a transition proba-
bility matrix for the unit time period which would be reasonable for a
severe chronic disease. The relative magnitude of the Aij
elements
covered a fairly wide range and included a zero value. The only effect
of the magnitude of the Aij which could be discerned in the study was
that of the small values. Invalid negative estimates of such Aij
occurred for even the largest sample size studied of ZOO. The relative
frequency of negative values of Aij for fixed i was inversely propor
tional to the magnitude of the Aij •. In addition, marginal distributions
of estimates of the zero-valued A23 , and of the PZ3 (t I) for the three
time periods approached the asymptotic normal distributions more slowly
98
than did the marginal distributions of the estimates of the other
parameters. The asymptotic normal distribution with mean zero is not
achievable for the modified non-negative estimate ~23 for any finite
sample size. The effect of the zero-valued parameter may have
extended to the joint distribution of the ~i' since the sample sizesJ .
for which it was acceptably normal for the unmodified estimates
paralleled the sample sizes for which the marginal distribution of
the unmodified ~23 was acceptably normal.
Any variation of the values of the A matrix causes a change
in the values of the p(t) matrix for given t. Such a change would not
necessarily alter the properties of estimates in samples generated
under the p(t'). Changes in the relative~agnitudeof A12 and A2l
would be most likely to have an effect on the properties of the esti-
mates. Changes in them, through the changes in P12(t') and P2l(t'),
would affect the relative. magnitude of the effective sample sizes, nl •
and n2•• These constitute an important factor in the properties of
the estimates. •
It is believed that the results of this Monte Carlo study apply
to a wide range of values of the A matrix provided it contains some
zero or near zero values. For A matrices containing no very small
values the properties of the estimates would be expected to be better
than observed in the study. However, to obtain firm conclusions for
processes with a radically different A matrix further investigation
would be necessary.
5.1.1.4 Number of Observations Per Patient, r, and Length of.. -!
Observation Time Period, t'. Small values of r equal 3, 5, 7 were
chosen for study because a number of follow-up studies in the past have
99
been based on as few observa(;ions. If the number of observations per
patient were larger for the same t' the distributions of the estimates
might be acceptably approximated by the asymptotic distributions for
somewhat smaller smaple sizes than indicated by the results of this
study.
Three values of r were studied for fixed t', and three values
of t' for fixed r, in an effort to isolate the effect of each of these
parameters on the properties of the estimates. For fixed t', the
sample sizes at which the marginal distributions of both the Pij and
~ij were acceptably approximated by the asymptotic normal distribu
tions decreased with increasing r. No effect of r was seen in the
approach of the joint distribution of the Pij to the asymptotic
distribution. The sample size at which the joint distribution of the
unmodified t .. was acceptably approximated by the asymptotic multivari-1.J
ate normal distribution decreased with increasing r.
For fixed r, t' had a marked effect on the frequency of
incomplete and divergent samples. For the shortest time period, very
few divergent samples but many incomplete samples occurred for the
small sample size of 10. For the longest time period, the results for
the same N were reversed, and for the intermediate time period a
moderate number of samples of each type occurred. The only detectable
effect of the time period on the properties of the estimates was the
somewhat larger sample size required for the marginal distributions of
the Pij for the shortest time period to be acceptably normal. This was
probably a result of the small values of the parameters Pij(t'), i # j,
for that t'.
100
It is apparent that the interrelation of the effects of rand
t' on the properties of the estimates is complex and would require
investigation of the sampling distribution for numerous combinations
of rand t' for adequate evaluation. This study was primarily Concerned
with the finite sa.ple size, if any, for which the asymptotic distribu
tions would provide adequate approximations. Hence, a wider range of
N was investigated for relatively narrow ranges of rand t'. Conse
quently, the results of th~ study extend only to combinations of rand
t' in the neighborhood of those investigated.
5.1.1.5 Sample Size. The range of N from 5 to 200, chosen
to ensure adequate investigation of sample size, was wider than would
normally be encountered in follow-up studies. Since it was necessary
to space rather widely the values of N for which samples were generated,
conclusions about the properties of the estimates are for approximate
intervals of sample size.
The intuitive feeling that sample size has a greater effect on
the properties of the estimates than does the number of observations
was confirmed. The probability 2!~ incomplete sample decreases more
rapidly with !than with~. Moreover, this probability tends to zero
with N for fixed r, whereas for fixed N and increasing r the limit is
greater than zero. N enters the expected value of effective sample
sizes as a factor. On the other hand, r determines the number of terms,
all less than one and of decreasing magnitude, which constitute the
other factor. The difference between the effects of Nand r rests on
the fact that the information contributed by different members of the
sample is independent, whereas the amount of information contributed by
101
a single individual during a study is limited by the probability of his
entering an absorbing state.
5.1.2 Considerations in Sample Design,
The occurrence of incomplete and divergent samples is an aspect
of the Markov process model peculiar to small sa~ple size. Since the
probability of such samples tends to zero with increasing II, they are
not considered in the asymptotic theory of the maximum likelihood
estimates. In application to follow-up studies the sample size is
always finite, and frequently severely limited, so that the probability
of a sample which does not yield estimates is a serious problem. If
such a sample does occur, one remedy is to add more patients to the
study. Another, perhaps less effective, is to continue observation on
the surviving patients of the original sample. Both of these, either
alone or in combination, constitute a sequential procedure, and further
investigation would be necessary to determine stopping rules and
properties of the estimates under such a procedure.
This investigation yielded some information which may be used
in designing a sample to reduce the probability of an undefined
estimate. In follow-up studies, it is customary and generally
necessary to select the sample from a population of patients in a state
of active disease. However, if the clinician had access to patients in
other transient states of the model who were comparable with respect to
prior treatment and other factors affecting the treatment response,
stratified sampling from all transient states would ensure a complete
sample. This is particula~ly important for a process with a high proba-
bility of absorption, since it could not be expected that continuing the
102
observations on the sample would ensure some patients entering all the
defined transient states.
To reduce the probability of a divergent sample, Zahl recom-
mended using prior informat~on on the relative frequency of transitions
out of transient states to choose an observational time period short
enough that appreciably less than one-half of the patients would be
expected to leave any transient state. The Monte Carlo results showed
that this does substantially reduce the probability of a divergent
sample. However, if the number of observations per patient is fixed,
the decrease in probability of a divergent sample is offset by an
increased probability of an incomplete sample, at least for the small
sample sizes of 10 or 20. Thus if r is fixed the choice of a small t'
requires an N large enough that the probability of an incomplete sample
is small.
In some cases a follow-up study may be planned in terms of the
total time period that the patients will be kept under observation.
Since the probability of a patient entering an absorbing state increases
with time, and no information is contributed after he does, the
expected amount of information per observation decreases with time.
The total~ period should ~ chosen short enough ~~ substantial
port~on ~ !!!!. sample may be expected !.2. survive lli period. For a
fixed total time period T = rt', and fixed N, an increase in the
frequency of observation decreases the probability of both an incomplete
and a divergent sample. It also increases the expected values of the
effective sample sizes of the Pij(t i), i = 1,2. However, the decrease
in the parameters p.. (t'), i # j, with decreasing t' necessitates'-J
larger effective sample sizes for comparable precision of their
103
estimates. Table 5.1 gives an example of the effects of varying rand
t' for approximately constant T for the values of A used in the MOnte
Carlo study.
Table 5.1. Expected effective sample sizes, probability of an
incomplete sample, and transition probabilities for
selected values of rand t'
T t' r E(nL
) E(n2• ) Pr[Inc.]Pij (t'),
i = 1,2 j = 1, ••• ,4
5.0 0.2 25 l3.04N 2.76N (.64)N .926 .029 .035 .010
.085 .894 .002 .019
5.0 1.0 5 2.95N 0.47N (.75)· .701 .100 .151 .048
.295 .588 .031 .086
6.0 2.0 3 1. SON 0.25N (.80)N .522 .129 .259 .090
.380 .376 .094 .150
In some chronic diseases an observation to determine the patient's
state involves laboratory work, such as biopsy, bone marrow samples,
blood tests, and X-rays, as well as a physical examination. The fre
quency of observation which is feasible in these cases is restricted
by the effect of the tests on the patient and the magnitude of the
laboratory work. If t' is limited in this way, N must be increased to
reduce the probability of undefined estimates. At times °it may also be
feasible to increase r to achieve the same result.
It is apparent that some prior knowledge of the magnitude of the
transition probabilities is essential to efficient sample design. If
the model with two transient states is used, crude estimates of
104
Pll(t) and P12(t) for some t would provide an estimate of the probabil
ity of an incomplete sample for that t as a function of rand N.
Values of the latter could then be determined to reduce the estimated
probability to an acceptable level. If, in addition, some estimates
of P21(t) and P22(t) were available, the A. ij , (i,j = 1,2) could be
estimated since they are not functions of the individual probabilities
of transition to the absorbing state's. From these estimates the
PiJ(t), (i,j = 1,2) could then be estimated for any t of interest and
the t' determined for which the Pii(t'), (i = 1,2) would be substanti
ally greater than 1/2. The probability of an incomplete sample and the
expected effective sample sizes could also be estimated for the chosen
t', and rand N selected accordingly. If the total number of observa-
tions made in a follow-up study is in some sense a measure of the cost,
then the sum of the expected values of the effective sample sizes is a
measure of the expected cost of the study.
5.1.3 Statistical Inference from Zah1's Estimates
5.1.3.1 Sample Size Less than Twent~. If a combination of small
enough t' with large enough r is not used, the probabilities of
incomplete and divergent samples are substantial for sample size less
than 20. Only some of the marginal distributions of the P1j(t') are
acceptably normal. The joint distributions of both the p.. (t l) and
, 1J
t ij are not acceptably approximated by the asymptotic multivaria~e
normal distributions. Use of Zahl's estimates is not recommended for
sample sizes less than 20 except in'a preliminary experiment to obtain
estimates of the Pij(t') ~or selection of optimal t' and r with a
larger sample.
105
5.1.3 .2 •Samp1@ Size Twenty to F,ifty. For N in the neighborhood
of 20, the probability of incomplete and divergent samples becomes
small provided a reasonably good choice of t' and r has been achieved.
The estimates Plj(t') of the probabilities of transitions from the
initial state are marginally normally qistributed. They have means
Plj(t'), and the as~p~otic variances. A possible exception is a
slight positive bias of PI2(t'), the estimate of the probability of
transition to the other transient state. The magnitude of the bias
is negligible for N = 50. The covariances of Plj(t'), P2k(t') are
zero for all j, k. However the sample estimates of the variances of
where
QI = N(l + Pll{t') + Pll(2t') + ... + Pll([r-l]t'»).
Pll(bt') is obtained from [~(t,)]b, and hence is'a function of the
P~j(t') as well as the Plj(t'). Thus, the estimates of the variances
of the Plj(t') are not independent of the P2j(t'), and normal theory
confidence intervals on the Plj(t') or two sample tests would be rough
approximations. Only the use of the point estimates Plj(t') is
recommended for sample sizes in the interval 20 to 50.
5.1.3.3 Sample Size Fifty to Two Hundred. For sample size
around 50, the joint distribution of the set Pij(t'), i = 1,2,
j = 1, ••• ,4, iF j, is acceptably approximated by the asymptotic multi-
variate normal distribution with mean vector Pij(t') and,covariance
matrix
106
8ik Pij(t')[8 jm - Pkm(t')]
Qi
There is some evidence, howev~r, that the normal moments of order
greater than two do not hold. Since the denominators of the sample
estimates of the covariances involve powers of the Pij(t') up to r-l,
multivariate normal tests of hypotheses concerning the Pij(t') and
confidence intervals on their estimates may still be approximate for
this interval of sample size.
The asymptotic multivariate normal is not an acceptable approxi-
mation to the distribution of the modif~ed t ij for sample size 50.
Nevertheless, the marginal distribution of each t ij is acceptably
normal provided the parameterAij is not too small. The marginal
dist~ibutions hold for the modified tlJ
for parameter values closer to
zero than for the t 2j • The ~ series estimates involve a variable
number of powers of the Pij(t'), and the lack of joint normality of the
t ij is probably due to non-normal higher moments of the Pij(t').
The sample estimates of the variances of the t ij require the
derivative of the A expansion in P*(t') evaluated for the Pij(t')
values. Thus they involve even higher powers of the Pij(t') for a
given computational accuracy than does the ~ series. For sample sizes
around 50 a multivariate statistic based on the Pij(t') will provide a
better test of equality of the parameters of the Markov proeess model
for two samples than will one based on the t ij • Such a test is limited,
of course, to samples obtained for the same observational time period.
This sample size will yield point estimates of the Aij which are almost
unbiased provided the Aij are not very small, and are marginally dis
tributed with asymptotic variances. Because of the lack of knowledge
~.•
107
of the distribution of the variance estimates, confidence intervals on
the ~ij cannot be considered reliable.
5.1.3.4 Sample Size Two Hundred. For sample sizes around 200,
the asymptotic multivariate normal distribution of the modified ~ij as
well as that of the Pi~(t') is acceptable for small t. The distribu
tionof the asymptotic variance-eovariances of the ~ij is still
questionable. In cases of a two sample experiment in which the same
observational time period was not possible for both samples an approxi-
mate multivariate normal test might be used for this sample size.
However, if the same time period were used a multivariate test based on
the Pij(t') would be more exact. Normal theory confidence intervals on
the ~ij using the variance estimates would also be approximate but
should be fairly reliable for sample sizes of this order.
5.1.4 Indications for Further Research
The findings in this Monte Carlo study 'that the asymptotic dis-
tributions of the estimates are aeceptable approximations for sample
sizes ranging from 50 to 200, even for'small r, indicate that further
investigation of the small sample distributions is warranted. The
distribution for any model depends on a complex of the A parameters
and the specifications of the sample design (r, t ' , N, and the initial
condition). Hence, it would seem preferable first to investigate in
greater detail the 'simple model of this study. Specifically, the areas
in which more sampling is needed are:
(i) other values of the A matrix,
(ii) varied composition of the sample with respect to initial state,
(iii) various combinations of r andt' for a range of fixed T = rt'.
,"
108
The results of this study indicate that sample sizes in the range 50 to
200 should be investigated.
The approximate normality of the distribution of the estimates for
moderate sample size also justifies investigation of the sampling dis-
tributionof the estimates of the asymptotic variance-covariances. The
asymptotic variance-covariances of the p.. (t ') are relatively simplel,J
functions of the Pij(t') and readily 'computed for the sample values
Pij(t'). The asymptotic variance-covariances of the ~ij are complex
functions of the p•. (t') and estimation of them requires evaluation of1J _
the derivative with respect to each p.. (t') of the A series expansion1J
in p*( t ') for .the sample values Pi .'( t '). Investigation of the samplingJ .
distribution of· these estimates would require extensive computations
and considerable computer time, but· the expenditure seems warranted in
view of the potential use of the estimate~ in tests· of hypothesis and
confidence intervals on the estimates of the p~rameters of the model.
5.2 Conclusions
5.2.1 Sample Size Less than Twenty
The probability of samples for which the estimates are undefined
is so large for small r that it is not feasible to estimate parameters
of the Markov process model with samples of size less than 20.-,
5.2.2 Sample Size Twenty
The probability of incomplete and divergent samples is small but
is still not negligible. For all except the shortest time period
studied the marginal distributions of the Plj(t') and the t lj are
acceptably approximated by normal distributions with means Plj(t') and
"
109
~lj' respectively, and the asymptotic variances. There is SOme evidence
of a small positive bias of P12(t') and A12 for this sample size and
the variance of Al2 may be slightly greater than the asymptotic.
5.2.3 Sample Size Fifty
There is a small probability of a divergent sample for the
longer time periods studied. For the Pij(t') the multivariate normal
distribution with mean vector Pij(t') and the asymptotic covariance
matrix is acceptable, although the marginal distribution of P23(t')
deviates from the asymptotic normal. The positive bias of P12(t') is
negligible fo~ this sample size. The normal distributions with means
Aij and the asymptotic variances are acceptable for the marginal dis
tributions of the Aij with the exception of A23 , 'the estimate of the
zero-valued parameter. 'The conclusion holds for both modified andI
unmodified estimates. The positive bias of Al2 is negligible for this
sample size.
5.2.4 Sample Size Two Hundred
The marginal distribution of P23(t') is acceptably approxi
mated by the asympto~ic normal. The multivariate normal distribution
with,mean vector Aij and the asymptotic covariance matrix is acceptable
for the unmodified Aij for all time periods studied. The joint dis
tribution of the modified non-negative Ai' approaches the asymptotic, J
distribution somewhat more slowly than does that of the unmodified
estimates. For sample,size 200 it is only acceptably approximated by
the asymptptic for the shortest time period studied. The marginal dis
tribution of the modified non-negative A23 cannot be normal with mean
.•
110
zero for any sample size, but the unmodified t 23 is marginally normally
distributed with zero mean and asymptotic variance for sample size
200 •
6. LIST OF REFERENCES
Anderson,T. W. and L. A. Goodman. 1957. Statistical inference aboutMarkov chains. Annals of Mathematical Statistics 28: 89-110.
Anderson, T. W. 1958. Introduction to MUltivariate StatisticalI Analysis. John Wiley and Sons, New York.
Birnbaum, Z. W. 1952. Numerical tabulations of the distribution ofKolmogorov I s statistic for finite sample size. Journal of theAmerican Statistical Society 47: 425-441.
Boag, J. w. 1949. Maximum likelihood estimates of the proportion ofpatients cured by cancer therapy. Journal of the RoyalStatistical Society, Series B ll: 15-6~.
Cochran, W. G. 1934. The distribution of quadratic forms in a normalsystem with applications to the analysis of covariance.Proceedings of Cambridge Philosophical Society ~: 178.
David, F. N. and J. Neyman. 1938. Extension of the Markov theorem onleast squares. Statistical Research Memoirs 1: 105-116.
Doob, J. L. 1953. Stochastic Processes. John Wiley and Sons, NewYork.
Fix, Evelyn and Jerzy Neyman. 1951. A simple stochastic model ofrecovery, relapse, death and loss of patients. Human Biology23: 205-241.
Geary, R. C. 1947. Testing for normality. Biometrika~: 209-242.
Haynam, G. E. 1957. A Statistical Interpretive System for theIBM 650 ~gnetic Drum Calculator. File Number 6.0.017,650 Program Library, Case Institute of Technology. '
Rac, M., J. Kiefer and J. Wolfowitz. 1955. On tests of normality andother tests of goodness of fit based on distance methods.Annals of Mathematical Statistics ~: 189-211.
Mainland, D., L. Herrera, andM.I. Sutcliffe. 1956. Tables for useswith Binomial Samples. New York University College of Medicine,New York.
a6senblatt, J. 1962. Note on multivariate goodness-of-fit tests.Annals of Mathematical .Statistics 33: 807-810.
112
Samuelson1 P. A. 1941. Conditions that the roots of a polynomial beless than unity in absolute value. Annals of MathematicalStatistics 11: 360-364.
Simpson, Paul B. 1951.bution function.476--478.
Note on the estimation of a bivariate distriAnnals of Mathematical Statistics 22:
..
Whittaker1 E. T. and G. W. Watson. 1935. Modern Analysis. CambridgeUniversity Press, Cambridge.
Wilks, S. S. 1935. On the independence of k sets of normallydistributed statistical variableso Econometrica~: 309-326.
Zahl, Samuel. 1955. A Markov process model for follow-up studies.Human Biology 
7. APPENDICES
7.1 Asymptotic Covariance Matrix Theorem
The k(k-l) square matrix F with elements
•
(1.7)
is the inverse of the matrix G with elements
where Qi =
I OPi1(t') OPij(t')
Pi' (t') ol~ OAJ ~ 7~
r-l
I nb(O) I Pbi(at'),b a=O
, (1.4)
and 8jm is the Kronecker delta.
Proof: The rows (columns) of the matrices are indexedby~a
double subscript, so that the elements ,are indexed by a pair of double
subscripts. Greek letter subscripts are constant with respect to the
summations. All matrices used in'the proof are k(k-l) square matrices.
For convenience .of writing, the. time dependent notation is omitted and
Pij(t') is denoted simply p ..•f 1.J
Let J(r) be the matrix the determinant of which is the
Jacobian of the transformation from the Abm to the Pij' i.e., the
element in the ijth row and bmth column is
•
114
Then
G = .1' (~) X J(~) •
Similarly" letJ(~) be the matrix the determinant of which is the" P
Jacobian of the inverse transformation from the Pij to the Abm so that
its elements are
,
and denote the
Then
OAbmOPij
transpose by .1' (~). Define the matrix Z by, P
8ibZij"bm = Q
i[8jmPij - PijPbm]
Hence" the theorem can be written
J(~) Z .1' (~) .1' (2.) X J(~) =: 1;P P A '"
where 1 is the identity matrix.
It is well known that
.1' (~) ..1' (:2.) :: 1 = J(~p) J(~) ,P A '"
so that if it can be shown that
. Z X = 1,
the theorem follows. Hence, it remains to show that
:f: uv,
::: uv.
If
115
Now
B\ B Q [....!.... + :J!!]L ib i P.i p ..
bm,b?'m 1. . 1.J
\' z x =L ij, bm bm,uvbm,bt'm ;
For ij :: uv,= [0, b~ i,
1, b == i,
so that
= - Pij]
:: 1,
which concludes the proof.
7.2 Properties of [p*(t')]S and Bounds on the Remainder after
a Finite Number of Terms of the A Series
Bounds on the remainder matrix after m terms of (1.2) are
based on properties of [p*(t')]S for the particular model studied. In
this section, for ease of writing, the time dependent and estimation
notation is omitted, and P*(t') is denoted si.mply P*. The ijth element
of (P*)s. is denoted by p~. for s =1, and by p*(s) for s ~ 2. ItJ.J ij
should be noted that in general
For the particular model p* has the form
* * * *P11 P12 P13 P14 Pll-1 P12 P13 P14* * * *
P21 P22 P23 P24 P21 P22-1 P23 P24...0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
where Pij 2:0,I Pij :: 1. It may be seen that
j
116
( _1)i+j+1 I * IPij .'
i,j ... 1,2,
and
i,j,k,e :: 1,2. (7.2.1)
Completeness of P and convergence of the series in p* require
(i) P12:: P!2 > 0,
(ii) PIIP22 - P12P21 > 0,
so that
o < Pn < 1,
For the case P ... 1.22 '
so that
*(s)P2j ...
*(s)P1j ...
Since
it follows that
j == 1,2,3,4.
I *(s+l ) t .< I. *( s ) I. P1j - P1j ,
with strict ,inequality for P~j ;. O.
s 2: 1,
117
For the case P22 < 1, several lemmas are necessary to show that
there exists an m such that
/*(s+l)1
Pij s ~ m, i = 1,2, j :: 1,2,3,4,
with strict inequality for P~j ~ O. To obtain an inequality needed for
the lemmas, p* is partitioned
so that
Since
* * * *Pll P12 P13 P14
*' * * *
[~lP21 P22 P23 P24 = ,
0 0 0 0
0 0 0 0
(p*)s :: [~].*
/1 - Pll ! *!Pll! :: > P1 2'
and
it follow.s that
By the rule for determinants,
so that
*(s) *(s) *{s} *(s) [* * '* * 1so < Pll P22 - P12 PZl = PllP42 - P1ZP21. < 1.
(A.2.2)
In the proofs of the lemmas both matrix pre- and post-multiplication
s+lexpressions for the elements of (P*) are employed, i.e.,
and
118
(7.2.3)
*(s+l)Pij (7.2.4)
then
Lemma 1. If for any j, and some m,
for s > m.
Proof: Let s =m+l. Then, using (7.2.3)
*(m+l) '*(m+l) * * [*(m)]2 * '* [ '*(m)]2Plj P2j = PIlP21 Plj + P12P22 P2j _
[ '* '* '* *] *(m) *(m)+ Pl1P22 + P12P21 Plj P2j •
From (7.2.1) and the premise, this is a sum of non-positive terms.t
Hence,
*(m+1) *(m+l) < 0Plj P2j -'
and the proof follows by induction.
Lemma-2. There exists an m such that
Proof: Case 1. j = 1,2. From (7.2.1) it may be readily- seen
that the!lemma is satisfied for m • 1•• I
The following corollaries to the lemma for case 1, which are
needed in the proof of case 2, are readily proved using (7.2.1),
(7.2.3), and (7.2.4).
Corollary 1.
C **(m) *(m) 0, P21 ~ o ,Pll P21 *0, P21 .= o.
Corollary 2.*(m+l) *(m)
Pij Pij < 0, i,j = 1,2.
Case 2. j =3,4.
* *(a) If Plj :: 0, or P2j = 0, the lemma is satisfied for m :: 1.
** . * ( )(b) For Plj , P2j > 0, assume P2l > O. For \I1.2:, 1, using 7.2.4
dividing through by the positive quantity [p*2 J 2,. J
*(m+l) *(m+l) = [ * ]2{a (p~j)2 + b P~j + c1,Pl j P2j P2j P~j P~j)
'where
119
and
a ""
b .. =
c=
*(m) *(m)P12 P21 "
Let fm+l denote
*(m+l) *(m+l)Pl j . ·P2j
* *the quadratic in Plj/p~jwhich determines the sign of
• The roots of fm+l are
and at the point of inflect~on the value of fm+l is
[ .~(~) *(m)] 2 ~ [ *(m) *(m)] 2_P...;:l::,;;l;..·_P...;:2::;.:2::..-."......_..,..P-r;1;;.,;;;2:....-P...;;2::,;;1:....-_ > 0
*(q>.) *(m)-2PU PZl
the inequality following from corollary 1. Considerin~the roots for
m= 1, it may be readily seen that
* *
0 <-P12
< 1 <-P22
~ ---;-Pll P21
Thus, f 2 ~ 0, for
120
* * *prj<
-P12 -P22<
Plj
* -;r- , or -;r- * ,P2j Pu P2l P2j
and the lemma is satisfied for m = 2, unless
*-P22 ·-r-P21
In the latter case consider the roots for m ~ 2. From (7.2.2)
and using (7.2.3) and substituting this inequality in the denominator,
so that
*(m+l) *(m)-P12
>-P12
*(m+1 ) *(m), ,.
PH Pll
It can be shown in similar fashion that
*(m+l) <P21
*(m+l)-P22
Thus, as m increases the roots of the quadratic approach each other andi '
* *Pl j !P2j" Since
11m, [ *(m). *(~)m -. co Pn P22
121
Thus, for some m, either'
or
so that'*(m+l) '*(m+1) < 0
Plj P2j -,
and the lemma is satisfied for that m+l for this case.
the quadratic in the expression ior
'* '*linear function of Pl j !P2j and it can be
( ) '* '* '*c For Plj ' P2j > 0, and P2l = 0,*(m+l) *(m+l)
Pl j P2j degeperates to a
shown in a similar manner that
'*(m+l) *(m+l)Plj P2j
'*< 0, for
Plj<*
P2j
with the root increasing with m so that the lemma is satisfied for some
m large enough.I
Corollary 3. For s > m , m satisfying lemma 2- 0 0 ;
I *(s+l) IPij :
Proof:
and
*(s) *(s)Pl j P2j < 0, by the premise,
so that!
,*($+1)Thus, Pij . is ~h~ sum of two terms of, like signs as required for the
,absolute value of a sum to be equa],lto the sum of the absolute values.
Lemma 3. If for some finite m > m of lemma 2o
*(m)Pij = 0,
122
then
*(8)Pij = 0, for s > m.
Proof: Case,l. j =1,2. Since by definition of convergent P*,
it can readily be shown that
~ts) *P2l = 0, if and only, if P2l = 0,
*(s) *(s) *(s)and Pu ' P12 ' P22 ' are non-zero except in the limit.
Case 2. j = 3,4. Let s ;:: m+l. Since m > mo' corollary 3 to
lemma 2 holds,
/.' *(m}/.Pl j=
and p*(m) zero implies bothlj
Theretore,
*(m) ° and *(m+l) °P2j =' , Plj:= ,'.
In the same manner it can be shown that zero p*(s) for any s > m implieslj*(s+l)zero Plj , so that the lemma follows by induction for all s.
Similarly,
*(m-l)If Plj"
then
zero,
By induction ~t can be shown that p~3m) zero implies p~3s} zero for
all s > m.
Lemma 4. There exists an ml such that
123
for all i,j,
*(ml)with strict inequality for Pij ~ O.
Proof: By definition (1.2) converges elementwise so that
00
Is=l
converges for all i,j.
Therefore,
so that
lims -. 00
lims -. 00
< 1,
for all i,j 0
That is, for every i,j, there exists some mij such that
Let
then
I *(s+1) I <Pij
m l1li
/*(s) /
Pij ,
maxi,j mij ,
for s > mOJo- 1
If
I *(s+l)/ <Pij /
*(s) /Pij , S ~ m, for all i,j.
... 0
the lemma is satisfied for that ij. For any non-zero term suppose.
I *(m+l) /Pij
III / *(m)/Pij •
Then for i • I"
by corollary 3 to lemma 2" and solving
124
b - I, ptll] ,p;~m)I
*P12
Therefore"
I *(m+1)I = * I *(m),P2j P21 P1j
so that
I *(m+2), =Pl j
Writing the bracketed coefficient in terms of the Pij"
~t can be seen that
Similarly" for i = 2, it can be shown that
I *(m+2), < ,*(m)1 = ,*(m+l),P2j P2j P2j •
Thus strict inequality holds for m1 = m+l" and the lemma is satisfied.
Lemma 5. If for some m > m , ml , where m satisfies lemma 2o 0 . .
and ml satisfies lemma 4,
*(m+l) *(m)IPij I S vlPij I" 0 < v < I" for all i,j,
with strict inequa~ity for p:jm) ~ 0, then
Ip:jS+l) , s Vlp:j~)1 ' for s ~ m, for all i,j,
*(m)with strict inequality for Pij ~ O.
Proof: The proof by induction follows readily using corollary
3 to lemma 2.
Theorem. Denote the matrix of the remainder after m terms of
the series (1.2) by
and define
125
*(m-l)0, p == 0,ij .
1*(m),
Pi~ *(m-l)* m-l) , Pij ~ 0,1P1j, I
) max ()v(m == i,j vij m •
Let m be greater than mo of lemma 2 and ml of lemma 4, then
*(m-l) .I.Pij T O.
*(m-l)= 0 P == 0, ij ,
( )1 *(m-1),/< v m Pij
1 - v(m) ,
==
Proof: From the definition of (1.2)
~ (_l)s+l *(s)I s Pij
s=tn+1
Thus
<00 , *(S)I\' PijL s •
s=m.fr1
so that
Case 1. P:3m-1) == O. By lemma 3,
*(s)Pij == 0, for s ~ m,
Case 2. p~~m-l) . /:..1J
I *(s)jPij·
o. By lemma 5,
*(s-l)<v(m)jPij I, for s > m,
126
and by $uccessive application,
Thus
Ip:js)1 < [v(m)]S-m+lIP:jm-l), , for s > ml
00 [ (.)]s-m+ll *(m-:-l)II vms Pij. .
s=m.+l
Let
00
Is=m+l
Then -W is the remain4er after·m terms of the Maclaurin seriesm
exp81lsion of log [1 - v(m)]. One form of the remainder for a Maclaurine. ..
series expansion of a function f(x) is (Whittaker and Watson, 193~)
xm(1 - e)~~f(m)(ex) 0 < e < 1
(m-l)! ' ,
where f(m)(ex) denotes the mth derivative evaluated at ex. Evaluating
this form of Wm yie.lds
Wm
Since
o < v(m) < 1, and 0 < e < 1,
both
o < ev(m) < e < 1, and 0 < ev(m) < vein) < 1,
so that
1 - e1 - ev(m) < 1,
Hence,
and
1 1and 1 - @V(m) < 1 - v(m}
127
, ~(m-1)IW< PJ.j m
[ v(m)]m-l< I *(m-1), v(m)
Pij 1 - v(m) •
G b,a.j;P(t)]
r J.
probability of a vector value
o~er the set of vector values
7.3 Eva1uatio! of the Probability Generating Function
7.3.1 Method of Evalqation
For the particular model studied the probability generating
function of the transitions, {nij }, py a single individau1 during r time
periods of length t, given t~at he was initially in 81, is
\' nllk n12k n14k n2lk= ~ ak8l1 8,12 ••• 814 e21k
where the coefficient ~ is the probability that the vector {nij} takes
the values {nijk}, and the summation is
{nijk} admissible under the model. The
{nijk} has the form
n11k n14k n21k n24k~ =ckP11(t) ••• P14(t) 'P21(t) •••P24(t) ,
where ~ is the number of paths an individual may traverse leading to
the same vector. of transitions. The generating function for N indi-
viduals is
The generating function was evaluated only for P(I.O), the"
transition matrix used- in generating the major portion of the Monte
Carlo samples. For brevity in the following discussion the generating
128
function is denoted simply G (N). For r sma~l the set of admissibler
transitions by an individual can be enumerated and the associated proba-
bility cgefficients evaluated for particular values Pij(t). For example;
G3 (1) is the sum of 20 terms, GS(l) of 66 terms, and G7(1) of 155 terms.
It was desired to evalu,te Gr(N) to at least N= 5 by macbi~e compu.ta
tion of successive powers of G (1). A 'squaring' and a 'G (1) multiplier'r r
program were used to obtain the powers of G3
(1) as follows:
G3
( 2) := [ G3
( 1 )] 2,
G3
(3) = G3
(1 ) •G3
( 2) ,
G3
(4) = [G3
(2)] 2 ,
G3 (S) = G3
(1) •G3
(4) •
Due to the memory limitation of the IBM 650, computation of
each-power had to.be carried out in two stages. Termwise cross products
were obtained in the first stage. Since several combinations of terms
could yield the same exponents, the output of stage 1 was then processed.-
to combine the coefficients of like cross products. The output of stage
2 consisted of
(i) identification - r, N, stage of computation,
(ii) (nijk,}'
(iii) ni • '
(iv) ~,
(v) ak - 10 decimal and rounded 6 decimal value.
It was feasible to program the computations only in single precision
arithmetic so that 10 decimal places were the maximum number that could
be retained for the probability coefficients.
129
The probability distribution of the Pij was obtained in a third
stage of the computations. Each sample point (nijk) was tested to
determine the type of sample, inco~plete, divergent, or convergent and
the estimate o,utput card was so coded. Foriricomplete samples only the
Plj were cOmputed.
The third stage ,program also cumulated the quantities
separately' for the three types of samples. Marginal 'probability distri-
butions were cumulated for each of the Pij in intervals of .01 for each
type of sample. At the completion of the program these results were
combined to yield marginal probability distributions, conditional mean
vectors and covariance matrices for
(i) the space of convergent samples,
(ii) the space of, complete samples, convergent and divergent,
(iii) the space of all samples for the Pl j '
7.3.2 Accuracy of the Evaluation and Magnitude of the
Computations
- . -10The probability coefficients of G3
(1) were accurate to 10
and summed to one. It can be shown that the error in the coeffici~nts
( ) -10for G3
2 was at most 10 and substantially less for almost" all of
them. Beyond the second power each coefficient was the sum of a
variable number of cross products and could be minus a variable number
of contributions les~ th.an 10-10• The sums of the probability,coeffi-
cients over the sample points as given in Table 7.3.1 give a rough
measure of the accuracy of the evaluation of the generating function.
130
Table 7.3.1. ~gnitude o~ output and total probabilities of probability
generating function evaluations for t equal 1.0
Probability Salllple points with Sample points with
generating - b l' > 10-10 -6proabi l.ty _ probability ~ 10
function Number Pr[setJ Number Pr[~etJ
G3 (1) 20 1.000 000 0000 20 1.000 000
G3 (2) 197 0.999 999 9998 197 0.999 997
G3 (3) 1142 0.999 999 9993 1020 0.999 976
G3,(4 ) 4850 0.999 999 9988 3112 0.999 870
G3 (5) 15182 . 1.000 001 8853 6864 0~999 584
.,
G5 ~,1) 66 1.000 000 0000 66 1.000 000
G5(2) 1070 c> .• 999 999 999~. 990 0.999 991
-.G7(1) 155 1.000 000 0000 155 1.000 000
131
The accuracy of the individual coefficients, of course, may be less than
that of the whole. The accuracy of the computations in stage 2 was
probability greater than
judged to be no greater -6than 10 and only sample points with
10-6 were included iIi the computation of·the
The number of terms from Table 7.3.1 provides some indicat~on of
the magnitude of the' computations. On the basis of the number of'terms
in GS
(2) and the progression of the terms for G3
(N) it was decided that
the further evaluation of GS(N) and G7(N) would require a prohibitive
amount of computation.
7.4 Monte Carlo Par~eter Tables
r/
7.4.1 ' Monte Carlo generatingmatrtc:es,
-.39 .16 .18 .05
.47 -.57 .00 .10A =
0 0 0 0
0 0 0 0
.926 339 39 .029 086 ~9 .034 648 67 .009 924 95
.085 443 05 .893 616 50 .001 587 95 .019352 50P(0.2) =
0 0 1.000 000 00 0
0 0 0 1.000 000 00
.701 202 04 .100 385 59 .150 673 35 .047 739 02
.294 882 70 .588 268 20 .0~1 118 58 .085 730 49p(l..O) :;:
0 0 1.000 000 00 0
0 0 0 1.000 000 00
133
e e
7 .4.2 Asymptotic covariance of (~ij (l.0)} , and of ~o~ified {tij) for r equal 3
e
'12 '13 '14 '21
.040 633 88 -.006 805 63 -.002 156 28 0
.057 580 01 -.003 236 46 01Ii I .020 454 S5 0
symmetric .904 669 29
t 121:
13 1:14 A2l
.146149 03 .003 214 24 -.008 236 83 .287 074 63
.102 248 24 .000 094 78 .054 803 32
11 .035 178 35 .008 647 17N
symmetric 3.614 448 30
'23 ' '24
0 0 I '12
0 0 I '13
0 0 '14
-.039 926 63 -.109 996 33 '21
.131,184 94 - .011 607 77 ,23
.341 038 28 ' '24
l23 1:24
-.006 092 98 .047 326 69~ t 12
-.042 388 06 - .000 511 04 I t l3
-.000 514 43 - .053 747 39 I >::14
-.321 970 09 -.057 23891 I l21
.266 298 91 .002 512' 02 I l23
.652, 770 70 l l24
....\IJ~
·e e
7.•4.3. Asymptotic covariance of (ll.•j
(l.O)}, andot'modified at,} 'for r equalS~ , J
e
1N
1N
'12
.030 643 89
symmetric
t 12
.103 907 80
symmetric
'13
-.005 132 35
.043 421 26
t 13
.002 187 ·22
.076 652 28
'14
-.001 626 19
- .002 440 67
.015 425 42
....),.14
-.005 285 30
.000 064 54
.025 632 11
fS21
o
o
o
.438 261 56
t 2'l'
.147 523 55
.033 252 03
.005 922 70
1.755 170 12
'23
o
o
o
-.019 342 38
.063 552 79
t23
-.003 337 11
- .026 835 94
-.000 308 12
-.157 602 32.
.130 472 13
'24
0 ~ '12
0 1'13
0'14
- .053 284 78 ~21
- .00.5 623 38 1123
.16.5 208 48 1124
t 24
.023 287 391 t 12
- .000 30677 I t13
- .027 983 93 I t 14
-.028 115 56 I t 21
.001 242 20 I t23
.316 662 01 I t 24 , ....wVI
e e~ ..
" ...... "'" '.~~
7.4.4. Asymptotic covariance of (p, .(l.0)}, and of modified (A,.} fer r equal 7. l.J l.J
e
1N
!N
~12
.026 422 16
symmetric
t 12
.087 966 42
s~etric
'13
-.004 425 36
.037~441 38
1:13
.001 825 11
.06.5 998 44
~14
-.001 402 12
-.002 104 51
.013 300 57
ti4
-.004 318 23
,.000 053 87
.021 870 04
~21
o
o
o
.314 995 46
...iA21
.109 42374
.026 590 37
.004 952 75
1.263 205 83
~23
o
o
o
-.013 901 53
.045 675 58
1:23
-.002 553 63
-.021 817 18
-.o~ 245 12
-.113 928 82
.094 358 82
~24'
0 l ~120 I '13
o I'14
-.038 298 19 . '21
-.004 041 56 I P23
.118 741 6!J '24
~24
.016 882 2tl ~12....
-.000 244 28 I A13
-.020 894 48 I t 14
-.020 364 65 I t21
.000 903 30 I t23
....•227 774 33 I A24 ....w---l 0\
e e
7.4.5. Asymptotic covariance of (Pij(0.2)}, and of modified Glij ) f~r r equal 7
e
symmetric
symmetric
!.Ii
!.Ii
~12
.004 957 31
t 12
.156 651 43
'13
-.000 176 91
.005 871 36
t1i
.000 139 86
.164 450 10
~l4
-.000 050' 68
-.000 060 36
.001 724 90
t 14
-.001 775 06
.QOO 002 63
.047 392 97
~21
o
o
o
.172 572 22
1:21
.096 993 55
.009 938 09
.002 428 59
5.733 807 15
~23
o
o
o
-.000 299 64
.003 501 30
t23
-.000 115 86
-.009 287 78
-.000 018 41
-.104 152 89
.100 388 16
'24
0 -, ~12
0 I ~13
0 I ~14
-.003 651 72 I '21
-.000 067 87 I P23
.041 91152 I P24
t 24
.018 689 16l t 12
.000 167 83 I t 13A
-.020 937 35 I A14
-.~26 789 62 I t 21
.000 173 08 I t23
1.197 049 50 I t 24 .....w'I
e e
7.4.6. ASyinptotic covariance of ('ij(2.0)J, and of "'modified rtij ) for r equal 7
e
!N
!N
'12
.047 764 42
symmetric
t I2
.147 718 51
symmetric
'13-.014 23.5 12
.081 439 10
t 13
.010 692 99
.065 133 63
'14
-.004 928 11
-.009 877 58
,.034 651 72
t 14
-.010 953 82
.600 288 22
.024 821 97
'21o
o
o
.513 757 87
t21
.
.299 657 15
.082 825 26
.006 90723
1.900 477 80
'23 ,'24
0 0 l'12
0 0 1P13
0 0'14
-.077 803 11 -.124 544 19'21
.185 410 24 -.030 741 36'23
.278 329 28J P24
t23 t 24
-.021 737 59 .033 965 79~t12
-.053 413 47- -.003 261 35 I t13
-.001 141 31 -.040 303 63 I 1:14
~
-.302 380 48 -.027 892 95 I l21~
.202 160 63 .004 .698 07 I l23
.245 896 92 I t 24....U)
ClO
7.4.7. Probability of an incomplete sample [h(r,~)]N
t l 0.2 1.0 2.0
.~ 7 3 5 7 7
1 .855 .829 .745 .704 .735
2 .730 .688 .555 .496 .540
3 .624 .570 .414 .349 .397
4 .534 .473 .308 .246 .292
5 .456 .392 .230 .173 .214
6 .390 .325 .171 .122 .158
7 .333 .270 .128 .086 .116
8 .285 .224 .095 .060 .085
9 .243 .185 .071 .042 .062
10 .208 .154 .053 .030 .046
15 .095 .060 .012 .005 .010
20 .043 .024 * * *,
25 .020 .009
30 .009 *
35 * ,
139
7.4.8. i Expected effective sample sizes by t' and r
140
t' r E(n1• ) E(n2•)
0.2 7 5.696 828 N .452 811 N
1.0 3 2.222 488 N .229 830 N
5 2.947 146 N .474 412 N
7 3.417 902 N .660 095 N
2.0 7 2.359 254 N .458 695 N
7.5 Probability Generating Function Evaluation_Tables
7.5.1. Means, variances, and covariances'of effective sample sizes
nl ., n2., determined from probability generating function
eva~uation for t' equal 1.0, by r,.
142
r 3 5 7Parameter
;
Mean n1• 2.222 488N 2.947 146N 3.417 900N
n2• 0.229 83ON 0.474 412N 0.,.666 246N --
Variance nl • 0,711 379N 2.277 020N 4.157 371N
e 0.295 115N 0.895 615N 1.570 095Nn2•
Covariance -0.180 972N -0.281 236N -0.093 453N
e e
7.5.2. Bias of (Pij(I.O)} for r equal 3, by sample type and sample size
e
'",.. '" . ",' '" '"'P12 . P13 P14 P21 P23 P24
Sample Type N Bias ~ Bias ~ Bias ~. Bias ~ Bias ~ Bias ~
Convergent 1 .39961 398 - .15067 -100 -.04774 -100 -.29488 -100 -.03112 -100 -.08573 -100
2 .18287 182 -.06308 -42 -.01999 -42 -.24143 -82 -.02072 -66 -.05705 -66
3 .12523 125 -.02684 -18 -.00846 -18 -.19638 -66 -.01843 -59 -.05074 -59
4 .08903 89 -.01258 -8 -.00392 -8 -.17084 -58 -.01646 -53 . -.04532 -53
5 .06663 66 -.00773 -5 -.00247 -5 -.14997 -51 -.01482 -48 -.04071 -48
Complete 1 .61555 .613 -.13761 -91 -.04360 -91 .05099 17 -.00377 -12 -.01037 -12
2 .22727 226 -.02701 -18 -.00857 -18 .04880 17 .00514 17 .01417 17
3 .12841 128 -.00991 -6 -.00311 -6 .04666 16 .00490 16 .01354 16
4 .08388 84 -.00434- -3 -.00131 -3 .04459 15 .00466 15 .01291 15
5 .05860 58 -.00238 -2 ';.00082 -2 .04258 14 .00438 14 .01228 14
All 1 .03816 38 .07951 53 .02519 53
2 .01784 18 ~03481 23 .01097 23
3 .01152 11 .02152 14 .00687 14
4 .00816 8 .01528 10 .00494 11
'.5 .00578 6 .01124 7 .00353 7 I-'
+:>IN
7.5.3. Convergent sample conditional covariance of (Pij(1.0)} for r equal 3,
by sample size
144
N... ,. ... ... ,. ...P12 PI3 P14 P21 P23 P24
1 0 0 0 0 0 0,.P12
2 .010 482 .000 558 .000 177 -.001 778 .000 838 .002 312
3 .on 028 -.000 490 -.000 157 .002 819 .000723 .001 998
4 .008 766 -.000 490 -.000 159 .003 633 .000 706 .001 950
5 .007 720 -.000 516 -.000 165 .003 985 .000 641 .001 790
1 0 0 0 0 0 ...P13
2 .017 698 -.002 431 -.002 444 -.000 230 -.000 636
3 .016 982 -.002 039 -.002 362 -.000 153 -.000 423
4 .014 569 -.001 364 -.001 859 -.000 140 -.000 392
5 .012 165 -.000 914 -.001 648 -.000 122 -.000 345
1 0 0 0 0,.P14
2 .007 268 -.000 777 -.000 073 -.000 201
3 .006 797 -.000 750 -.000 048 -.000 135
4 .005 578 -.000 591 -.000 048 -.000 126
5 .004 490 -.000 522 -.000 042 -.000 110
1 0 0 0 ...P21
2 .023 867 -.000 556 -.001 533
3 .035 522 -.001 020 -.002 813
4 .041 100 -.001 245 -.003 369
5 .044 961 -.001 437 -.003 971
1 0 0,.P23
2 .004 964 -.000 265
3 .005 643 -.000 315
4 .006 120 -.000 376
5 .006 350 -.000 430
1 0,.P24
2 .013 221
3 .015 009
4 .016 226
5 .017 086
7.S.4. Complete sample conditional covariance of (Pij(1.0)) for r equal 3,
by sample size
145
NA
P13A A A '"P12 P14 P21 P23 P24
1 .061 340 -.002 820 -.000 893 -.040494 .001 357 .003 737 ""P12
2 .024 589 .003 305 .001 047 -.007 283 Aooo 563 .001 554
3 .013 315 .000 982 .000 305 -.002 778 .000 114 .000 325
4 .009 272 .000 187 .000 052 -.001 841 .000 015 .000 059
5 .007 133 -.000 162 -.000 061 -.001 311 -.000 047 -.000 083
1 .006 359 -.000 054 .008 542 -.000 357 -.000 984 '"P13
2 .029 095 -.004 371 .000 547 .000 152 .000416
3 .021 558 -.002 078 .000 378 .000 105 .000 283
4 .016 191 -.001 248 .000 250 .000088 .000 229
5 .012 789 -.000 854 .000 268 .000 068 .000 175
1 .002 052 .002 706 -.000 113 -.000 312 ""P14
2 .012 193 .000 168 .000 047 .000 133
3 .008 250 .000 191 .000 032 .000 087
4 .006 010 .000 077 .000 023 .000 073
5 .004 640 .000 076 .000 015 .000 056
1 .200 751 -.009 460 -.026 063 ""P21
2 .190 671 -.011 890 -.032 765
3 .180 887 -.011 187 -.030 831
4 .171434 -.010 510 -.028 969
5 .158 737 -.009 856 -.027 198
1 .019 341 -.002 061 ""P23
2 .030 750 -.003 457
3 .029 080 -.003 252
4 .027 468 -.003 056
5 .025 900 -.002 870
1 .049 668 ""P24
2 .078 660
3 .074 437
4 .070 352
5 .066 438
7.5.5. All samples covariance of ($lJ(l.O)l for r equal 3
by sample size
N '12 '13....P14
1 .084 188 -.030 777 -.009 751 '122 .030 542 -.006 682 -.002 111
3 .017 746 -.003 254 -.001 036
4 .012 380 -.002 082 -.000 667
5 .009 384 -.001 546 -.000 498
1 .133 288 -.016 787 '13
.2 .047 988 -.003 125
3 .026 998 -.001 490
4 .018 545 -.000 957
5 .013 982 -.000 714
1 .053 699 '14
2 .017 266
3 .009 586
4 .006 565
5 .004 928
146
7.5.6. Convergent sample conditional mean product errors of (p .. (1.0))~J
for r equal 3, by sample size
147
N,.. ,.. ,.. ,.. ,..
"P12 P13 P14 P21 P23 P24
1 .159 688 -.060 209 -.019 077 -.117 837 -.012 436 -.034 259 "P12
2 .043 924 -.010 977 -.003 479 -.045 928 -.002 951 -.008 121
3 .026 703 -.003 845 -.001 221 -.021 770 -.001 581 -.004 349
4 .016 793 -.001 610 -.000 508 -.on 577 -.000 579 -.022 085
5 .on 640 -.001 031 .000 329 -.006 007 -.000 347 -.000 922.--
1 .022 701 .007 193 .01+4 430 .004 689 .012 917 "P13
2 .021 677 -.001 170 .012 786 .001 077 .002 962
3 .017 700 -.001 811 .002 901 .000 340 .000 936
4 .014 728 -.001 314 .000 290 .000 067 .000 178
5 .012 224 -.000 895 -.000 489 -.000 008 -.000 030
1 .002 279 .Ollf 078 .001 486 .004 093 "P14
2 .007 668 .004 051 .000 341 .000 939
3 .006 869 .000 919 .000 108 .000 296
4 .005 593 .000 079 .000 016 .000 052
5 .004 496 -.000 151 -.000 006 - .000 010
1 .086 954 .009 177 .025 280,..P21
2 .082 155 .004 446 .012 241
3 .074 095 .002 594 .007 144
4 .070 287 .001 567 .004 374
5 .067 452 .000 785 .002 135
1 .000 968 .002 668 "P23
2 .005 393 .000 917
3 .005 981 .000 618
4 .006 391 .000 370
5 i .006 569 .000 173I
1 .007 350,..P24
2 .016 475
3 .017 579
4 .018 280
5 .018 743
7.5.7. Complete sample conditional mean product errors of (Pij(l.O;)
for r equal 3, by sample size
148
.... ....P14
.... ....P24N P12 P13 P21 P23
1 .440 242 -.087 526 -.027 731 -.009 108 -.000 964 -.002 646 "..
P122 .076240 -.002 833 -.000 901 .003 808 .001 731 .004 774
3 .029 801 -.000 289 -.000 093 -.003 206 .000 744 .002 058
4 .016 308 -.000 177 -.000 058 -.001 900 .000 406 .001 142
5 .010 567 -.000 302 -.000 109 -.001 184 .000 210 .000 637
1 .025 295 . 005 946 .001 525 .000 162 .000 443 ....P13
2 .029 825 -.004 139 -.000 771 .000 013 .000 033
3 .021 656 -.002 048 -.000 083 .000 056 .000 149
4 .016 210 -.001 242 .000 056 .000 068 .000 173
5 .012 795 -.000 852 .000 167 .000 057 .000 146
1 .003 953 .000 483 .000 051 .000 140 ""P14
2 .012 267 -.000 250 .000 003 .000 Oll
3 .008 250 .000 177 .000 030 .000 083
4 .006 012 .000 018 .000 017 .000 056
5 .004 640 .000 041 .000 011 .000 045
.203 351 -.009 653 -.026 592 ""1 P21
2 .193 053 -.011 640 -.032 073
3 .183 059 -.010 959 -.030 202
4 .173 422 -.010 302 -.020 393
5 .160 550 .009 669 -.026 675
.019 356 -.002 022 ....1 P23
.2 .030 777 -.003 384
3 .029 104 -.003 185
4 .027 490 -.002 997
5 .025 920 -.002 816
•049 775 ....1 P24
2 .078 861 -
3 .074 619
4 .070 519
5 .066 589
7.5.8. All samples mean product errors of (Pj
(1.0) for
r equal 3, by sample size
N .~12A '"P13 P14
1 .085 645 -.027 743 -.008 790 "P12
2 .030 860 -.006 001 -.001 915.
3 .017 878 -.003 007 -.000 958,
4 .012 447 -.001 957 -.000 627
5 .009 422 -.001 481 -.000 478
"1 .148 400 -.014 784 "P13
2 .049 200 -.002 743
3 .027 460 -.001 343
4 .018 779 -.000 881
5 .014 109 -.000 674
1 .054 333 "P14
2 .017 387
3 .009 632
4 .006 589
5 .004 941
149
7.6 Monte Carlo Data Tables
7.6.1. 'Means, variance-s, a~d covariance Of effective sample,
sizes for t eq~a1 1.0, r equal 3, by sample size
151
Mean Variance Covariance
N n1. n2• n1. n2• n1.,n2•
5 Parameter . 11.11 1.15 3.56 1.48 -0.90
Samples 10.74 1.36 3.75 1.91 -1.13
11.06 1.30 3.20 1.40 -0.53
10 Parameter 22.22 2.30 7.11 2.95 -1.81
Samples 22.00 1.78 5.51 1.97 -0.80
22.66 2.22 5.86 2.71 -1.64
20 Parameter 44.45 4.60 14.23 5.90 -3.62
Samples 45.84 4.42 22.67 8.70 -8.60
44.20 5.02 9.76 5.45 -2.00
50 Parameter 111.'12 11.49 ' 35.57 14.76 -9.05
Samples 109.52 11.18 45.03 14.19 -5.42
1111.56 11.06 38.41 13.28 -7.50
100 Parameter 222.25 22.98 71.14 29.51 -18.10
Samples 220.22 22.56 55.11 42.90 -27.31
223.42 23.04 59.19 21.63 -7.41
200 Parameter 444.50 45.97 ' 142.28 59.02 -36.19
Samples 440.62 47.32 121.,63 54.47 -38.73
443.00 , , " , 148.45 62.01 -8.8246.48 ,
7.6.2. Means, variances, and covariance of effective sample
sizes for r equal 1.0, r equal 5, by sample size
152
Mean Variance Covariance
N
n1. n n1• n2• n1. ,n2•2•.
5 Parameter 14.74 2.37 11.38 4.48 -1.41
Samples 15.16 2.48 10.67 5.89 -1.22
14.16 2.10 12.79 2.95 -0.89,
10 Parameter 29.47 4.74 .22.77 8.96 -2.81
Samples 29.00 '4.68 22.24 9.32 -4.80
30.06 5.08 29.49 10.77 -1.64
20 Parameter 58.94 9.49 45J54 17.91 -5.62
Samples 58.02 9.94 39.33 28.59 -11.59
58.76 9.86 51.00 21.47 -10.89
,50 Parameter 147.36 23.72 113.85 44.78 -14.06
Samples 147.54 23.50 117.44 49.15 -1.21
146.70 24.12 92.50 66.35 -6.53
100 Parameter 294.71 47.44 227.70 89.56 -28.12
Samples 295.92 47.02 171.14 94.71 -0.02
296.00 46.32 209.88 113.57 -26.02
200 Parameter 589.43 94.88 455.40 179.12 -56.25
Samples 590.44 94.18 496.17 195.90 -107.86I
595.62 95.42 416.28 ,176.70 -27.35, '
7:6.3. Means, variances, a~dcovariance of effective sample
sizes for t! equal 1.0, r equal 7, by sample size
153
Mean Variance Covariance
NI
. n1. n2. n1. n2• nl. ,n2•
5 Parameter 17.09 3.33 20.79 7.85 -0.47
Samples 17.12 4.10 18.07 12.70 -0.18
17.90 2.72 20.70 6.57 +0.05
10 Parameter 34.18 6.66 41.57 15.70 -0.93
Samples 34.46 7.78 32.05 15.81 -4.63
33.74 7.68 35.46 13.36 1.43
20 Parameter 68.36 13.32 83.15 31.40 -1.87/
Samples 66.26 13.64 74.81 36.77 -2.11
68.22 13.12 80.09 32.92 -2.60. ....
50 Parameter,
170.90 33.31 207.87 78.50 -4.67
Samples 169.76 33.52 228.72 82.17 -7.73
175.50 32.76 179.89 62.59 17 .71
100 Parameter 341. 79 66.62 415.74 ,157.01 -9.34
Samples 344.48 66.40 302.21 119.80 55.54
340.48 67.30 347.03 215.85 ';'75.00l.,
200 Parameter 683.58 133.25 831.47 314.02 ...18.69
Samples 683.72 133.36 650.98 383.21 -29.88
684.76 132.58 632.15 312.82 -3.29'.
e e
7.6.4. Estimates of bias of (Pij(l.O») and modified (tij ) from convergent samples
in sets of 50 for r equal 3, by sample size
Parameter .100 .151 .048 .295 .031 .086 .16 .18 .05 .47 0 .10
N Wa ,., ,., ,., ,., ,., ,.,
~12 ~13 ~14 ~21 t23 ~24cn P12 P13 P14 P21 P23 P24
5 22 .072 -.005 -.007 -.101 -.031 -.070 .117 .028 .006 -.088 .000 -.081
22 .071 -.054 .066 .001 -.016 -.086 .169 -.031 .033 .160 .019 -.100
10 27 .014 -.003 .011 -.050 -.013 -.020 .079 .013 .021 .140 .026 .004
32 .022 -.009 .088 -.045 .004 -.055 .054 -.005 .016 .026 .054 -.055
20 44 .005 -.008 .000 -.045 .013 -.002 .015 -.008 .003 -.004 .057 .021
45 .008 -.001 .004 -.018 ' .011 .002 .034 .000 -.005 .097 .055 .029
50 48 .001 .003 .004 .003 -.001 .002 .008 .005 .006 .039 .029 .011
50 -.003 .005 -.004 -.033 -.011 .013 -.005 .008 -.006 -.039 .019 .029
100 50 -.003 .006 .002 .025 -.008 -.004 .001 .009 .003 .070 .015 .000
50 -.002 -.005 .004 .009 -.008 -.003 -.006 .005 .005 .031 .014 .000
200 50 .001 .005 .000 -.020 .005 .001 .001 .006 .000 -.027 .023 .002
50 .002 .002 -.001 -.003 -.003 -.008 .004 .005 -.001 -.001 .010 -.011
aW denotes the number of convergent samples.cn
e
....~
e e
7.6.5. Estimates of bias of (Pij(1.0)} and modified (tij } from convergent samples in sets
of 50 for r equalS, by sample size
Parameter .100 .151 .048 .295 .031 .086 .16 .18 .05 .47 0 .10
N Wa ,., ,., ,., ,., ,., ,., t 12 t 13 t 14 ~21 t
23t 24cn P12 P13 P14 P21 P23 P24
5 31 .042 -.008 -.015 -.046 -.015 -.016 .124 .016 -.014 .080 .022 .001
29 .026 .036 -.007 -.068 -.016 .036 .081 .071 .006 -.049 .018 .069
10 40 .022 -.010 .015 -.023 -.014 .025 .074 -.003 .021 .069 .024 .075
40 .013 -.008 .005 -.015 -.005 -.038 .034 -.004 -.002 .042 .029 -.035
20 50 .009 -.002 .003 -.027 .002 .035 .033 .001 .001 .020 .040 .071
45 .012 .002 -.003 .003 .009 .012 .036 .006 -.005 .072 .040 .041
50 50 -.001 -.001 .000 -.013 .004 .008 .004 .000 -.001 .000 .025 .017
50 .001 .002 .000 .026 -.008 .004 .010 .000 .000 .078 .012 .015
100 50 -.001 .001 .001 .002 .000 .004 .002 .002 -.001 .015 .013 .008
50 -.003 .003 -.003 -.002 .000 .006 -.003 .003 -.004 .005 .015 .011
200 50 .001 .000 .000 .003 .004 .004 .004 -.001 .001 .014 .013 .006
50 .002 .,..006 .000 .003 .000 .002 .004 -.007 -.001 .010 .011 .003
aW denotes the number of convergent samplescn
e
....VIVI
e e
7.6.6. Estimates of bias of (p .. (1.0)) and modified (ti .) from convergent samples inLJ J
sets of 50 for r equal 7, by sample size
Parameter .100 .151 .048 .295 .031 .086 .16 .18 .05 .47 0 .10
a'" '" '" '" '" '" ~12 t
13 ~14 ~21 ~23 ~24N W P12 P13 P14 P21 P23 P24cn
5 31 .047 .009 -.013 -.070 -.025 -.015 .114 .045 -.006 -.026 .007 .003
24 .032 -.015 .010 -.048 .005 -.005 .078 -.012 .022 -.024 .048 .001
10 46 .012 -.001 .000 -.010 .018 -.012 .030 .000 .003 .037 .063 .004
46 .014 -.010 .006 .014 -.018 .004 .034 -.007 .008 .091 .014 .024
20 50 .005 .010 .000 .018 -.011 .009 .018 .018 .000 .091 .018 .031
49 .003 -.007 .004 -.012 .012 .003 .022 -.009 .004 .063 .049 .022
50 50 .001 -.001 .003 -.002 .001 .004 .004 .000 .002 .020 .021 .008
50 -.002 -.005 .000 .017 -.003 .008 .002 -.006 -.001 .048 .015 .013
100 50 -.002 .000 -.002 -.008 -.001 .003 -.003 .000 -.003 -.011 -.013 .005
50 .002 .003 -.001 -.001 -.004 .009 .005 .004 -.003 .007 .008 .015
200 50 .000 .001 -.001 .004 .000 -.003 .001 .001 -.001 .011 .008 -.003
50 .002 .000 .000 .010 .003 .002 .005 .001 .000 .028 .008 .004
aW denotes the number of convergent samplesen
e
I-'1rI()\
-''-Je e
7.6.7. Estimates of bias of {Pi' .(0.2) and modified' tt.j
)J. 1
from convergent samples in sets of SO for r equal 7, by sample size
e
Parameter .029 .035 I .010 I .085 I .002 I .019 I .16
aI ~14 I ~21 I ~23 I ~24 I t 12N I W ~1~ ~13en
.18 I .05
t 13 I, t 14
.47
t 21
o
t ~
23
.10
t 24
10 I 37 I .003 .001 •• 003 1.042 -.002 .028 .025 -.008 -.018 .. 219 0 .043
40 I .005 .002 -.002 .001 .036 .012 .005 .021 0 .034.001 .016
50 I 50 I .001 I .001 I .000 I -.007 -.001 .004 .008 .005 -.003 -.025 .003 ..028
SO .001 .001 .000 .018 .000 -.001 .006 .004 -.003 .122 .013\ .003
200 I SO .000 .000 .000 .001 .000 .004 .000 -.002 ,- .001 .012 .0081" .025
50 I .000 I .000 I .000 I .0001 .001 I .~oo 1-.002 -.002 •• 001\ .0061:0111- .002
aW denotes the number of convergent samples.en .
....VI....,
e e
7.6.8. Estimates of bias of {p.. (2.0») and modified {ii,}l.J . J
from convergent samples in sets of 50 for r equal ~ by sample size
,.
Parameter .129 .259 •.090 .380 .094 .150 .16 .18 .05 .47 0 .10,I a
>:23 >:24N I 1-1 iS12 iS13 is14 iS21 is23 is24 t 12 t 13 t 14 t 21. en
10 31 .025 .026 -.001 -.050 -.002 .018 .071 .038 .006 .019 .054 .044
30 .019 -.008 .011 -.075 .012 -.002 .039 .001 .012 -.078 .070 .003,
50 9' -.001 .005 -.004 -.011 .002 .014 .005 .006 -.006 .012 .026 .0204 ::
49, .001 -.002 .004 -.008 .006 .017 .019 -.001 -.000 .043 .032 .026
'200 -50 -.002 .006 .002 -.006 .000 .008 .000 .005 .001 .005 .011 .008
50 -.002 .003 -.002 .002 .005 .009 .006 .003 - .003 .028 .012 .013
-aW denotes the number of convergent samples.cn
e
....V100
e e
7.6.9. Estimates of variance of (P1j(1.0») from convergent samples in sets of 50 for r equal 3, 5, 7,
by sample size
,.,"
,.,P12 P13 P14
N~3 5 7 3 5 7 3 5 7
5 Asymptotic .008 127 .006 129 .005 284 .011 516 .008 684 .007 488 .004 091 .003 085 .002 660
Samples .009 021 .008 478 .006 091 .016 287 .008 985 .008 643 .004 582 .002 275 .001 550
.005 199 .004 269 .004 715 .008 509 .011 318 .007 241 .004 639 .003 977 .005 434
10 Asymptotic .004 063 .003 064 .002 642 .005 758 .004 342 .003 744 .002 045 .001 542 .001 330
Samples .002 735 .002 963 .002 281 .005 427 .003 399 .002 208 .002 286 .003 017 .001 246
.003 315 .002 536 .001 905 .006 898 .003 847 .002 908 .002 308 .001 247 .001 100
20 Asymptotic .002 032 .001 532 .001 321 .002 879 .002 171 .001 872 .001 023 .000 771 .000 665
Samples .003 082 .002 436 .001 379 .003 232 .001 679 .001 476 .001 064 .001 188 .000 663
.001 645 .001 589 .001 064 .002 127 .002 113 .001 324 .000 866 .001 056 .001 071
50 Asymptotic .000 813 .000 613 .000 528 .001 152 .000 868 .000 749 .000 409 .000 308 .000 266
Samples .000 840 .000 772 .000 538 .001 275 .000 870 .000 819 .000 477 .000 415 .000 328
.000 653 .000 952 .000 494 .001 697 .000 772 .000 675 .000 386 .000 372 .000 321,
100 Asymptotic .000 406 .000 302 .000 264 .000 576 .000 434 .000 374 .000 204 .000 154 .000 133
Samples .000 704 .000 349 .000 201 .000 587 .000 431 .000 388 .000 179 .000 161 .000 118
.000 365 .000334 .000 274 .000 552 .000371 .000 270 .000 271 .000 118 .000 113
200 Asymptotic .000 203 .000 153 .000 132 .000 288 .000 217 .000 187 .000 102 .000077 .000 066
Samples .000 201 .000 171 .000 154 .000 227 .000 231 .000 160 .000 100 .000 078 .000 071
.000 151 .000 142 .000 121 .000357 .000 217 .000 194 .000 107 .000 080 .000 069
e
....In\0
e e
7.6.10 Estimates of variance of (P2o(1.0)} from convergent samples in sets of 50 for r equal 3, 5, 7, byJ .
sample size
,. ,. ,.P21 P P2423
~ 3 5 7 3 5 7 3 5 7
5 Asymptotic .180 940 .102 109 .062 999 .026 237 .012 711 .009 135 .068 208 .033 043 .023 748
Samples .048 321 .051 462 .034 181 0 .008 065 .000 643 .005 050 .019 489 .011 719
.058 021 .035 496 .031 877 .005 050 .003 411 .012 456 0 .031 283 .010 384
10 Asymptotic .090 470 .051 054 .031 500 .013 118 .006 355 .004 568 .034 104 .016 522 .011 874
Samples .077 261 .031 466 .026 700 .009 259 .002 794 .007 678 .018 109 .022 445 .013 160
.059 079 .035 271 .023 113 .009 981 .004 735 .001 249 .007 971 .008 191 .010 737
20 Asymptotic .045 235 .025 527 .015 750 .006 559 .003 178 .002 284 .017 579 .008 261 .005 937
Samples .044 461 .020 119 .021 037 .008 617 .004 142 .001 645 .016 201 .015 697 .009 098
.046 775 .019 157 .020 821 .008 016 .003 715 .004 585 .019 324 .012 070 .008 042
50 Asymptotic .018 094 .010 211 .006 300 .002 624 .001 271 .000 914 .006 821 .003 304 .002 375
Samples .014 893 .010 094 .008 122 .002 560 .001 340 .001 077 .007 268 .004 476 .002 321
.015 145 .008 872 .006 538 .002 182 .000 896 .000 783 .009 981 .005 862 .002 017
100 Asymptotic .009 047 .005 105 .003 150 .001 312 .000 636 .000 457 .003 410 .001 652 .001 187
.012 583 .004 371 .003 288 .001 219 .000 611 .000 646 .004 492 .002 034 .001 320
.006 933 .004 598 .003 149 .000 989 .000 730 .000 435 .003 570 .001 518 .001 962
200 Asymptotic .004 524 .002 553 .001 575 .000 656 .000 318 .000 228 .001 752 .000 826 .000 594
Samples .005 769 .002 627 .001 692 .000 959 .000 291 .000 214 .001 487 .000 496 .000 360
.004 303 .001 692 .001 891 .000 465 .000 309 .000 169 .001 275 .000 846 .000 521
e
.....S
e e
7.6.11. Estimates of covariance of (P1j(1.0),P1k(1.0)} from convergent samples in sets of 50 for r equal
3, 5, 7, by sample size
A. A. A. A. A. A.
P12' P13 P12' P14 P13,P14
N----------': 3 5 7 3 5 7 3 5 7
5 Asymptotic -.001 361 -.001 026 -.000 885 -.000 431 -.000325 -.000 280 -.000 647 -.000 488 -.000 421
Samples -.003 768 .000 909 .001 281 -.000 434 -.000 506 -.000 478 .000 978 -.000 824 -.000 699
.001 443 .000 826 -.002 438 .001 023 .000 936 .003 583 -.000 252 -.000 671 -.001 339
10 Asymptotic -.000 680 -.000 513 -.000 442 -.000 216 -.000 163 -.000 140 -.000 324 -.000 244 -.000 210
Samples -.000 734 .000 359 .000 163 -.000 018 .000 042 -.000 686 -.000 392 -.000 826 .000 342
-.000 891 .000 260 -.000 775 .000 352 .000 180 -.000 121 -.001 220 .005 127 -.000 211
20 Asymptotic -.000 340 -.000 257 -.000 222 -.000 108 -.000 081 -.000 070 -.000 162 -.000 122 -.000 105
Samples -.000 757 -.000 202 -.000 442 .000 390 .000 057 -.000 193 -.000 135 -.000 594 .000 301
-.000 571 .000011 -.000 203 -.000 083 .000 011 -.000 077 -.000 170 -.000 058 -.000 115
50 Asymptotic -.000 136 -.000 103 -.000 088 -.000 043 -.000 032 -.000 028 -.000 065 -.000 049 -.000 042
Samples -.000 179 -.000 245 .000 069 -.000 178 -.000 109 -.000 083 .000 180 .000013 -.000 114
-.000 157 -.000 109 -.000 149 .000 035 -.000 133 -.000 039 -.000 096 -.000 127 -.000 065
100 Asymptotic -.000 068 -.000 051 -.000 044 -.000 021 -.000 016 -.000 014 -.000 032 -.000 024 -.000 021
SCimples -.000 131 -.000 140 -.000 068 -.000 044 -.000 018 -.000 025 -.000 091 -.000 054 -.000 019
-.000 096 -.000 075 -.000 038 -.000 025 -.000 037 -.000 005 .000 029 -.000 053 -.000 043
200 Asymptotic -.000 034 -.000 026 -.000 022 -.000 011 -.000 008 -.000 007 -.000 016 -.000 012 -.000 010
Samples -.000 032 -.000 014 -.000 001 -.000 027 .000 002 -.000 024 -.000 040 -.000 021 -.000 027
- 000 067 .000 019 -.000 042 0 -.000 042 -.000 004 -.000 002 -.000 037 -.000 013
e
.....0".....
e e
7.6.12. Estimates of covariance of (P2j(1.0), P2k(1.0)} from convergent samples in sets of 50 for r
equal 3, 5, 7, by sample size
'" '" '" '" '" '"P21' P23 P21'P24 P23' P24
N________r 3 5 7 3 5 7 3 5 7
5 Asymptotic -.007 985 -.003 868 -.002 780 -.021 999 -.010 658 -.007 660 -.002 322 -.001 125 -.000 808
Samples 0 -.004 151 -.000 414 -.003 078 -.005 602 -.002 016 0 -.001 165 .000 210
-.004 690 -.001 408 -.006 374 0 -.017 343 -.005 192 0 -.000 535 -.001 315
10 Asymptotic -.003 993 -.001 934 -.001 390 -.011 000 -.005 329 -.003 830 -.001 161 -.000 562 -.000 404
Samples -.004 713 -.000 540 -.002 245 -.003 251 -.000 340 -.008 104 -.001 270 -.000 514 .000 492
-.000 401 .000 271 -.001 666 -.000 329 -.004 976 -.003 788 -.000 621 -.000 593 .000 699
20 Asymptotic -.000 996 -.000 967 -.000 695 -.005 500 -.002 664 -.001 915 -.000 580 -.000 281 -.000 202
Samples -.004 528 -.000 343 -.001 048 -.004 572 -.005 402 -.002 698 -.001 715 -.000 391 -.000 599
-.004 794 -.001 202 -.001 760 -.007 294 -.004 035 .000 332 .000 476 .000 047 -.000 451
50 Asymptotic -.000 798 -.000 387 -.000 278 -.002 200 -.001 066 -.000 766 -.000 232 -.000 112 -.000 081
Samples -.000 778 -.000 579 -.000 683 -.003 677 -.001 669 .000 046 -.000 524 -.000 071 .000 044
-.000 339 -.000 465 -.000 170 -.002 486 -.001 053 -.001 046 -.001 282 -.000 259 .000 047
100 Asymptotic -.000 399 -.000 193 -.000 139 -.001 100 -.000 533 -.000 383 -.000 116 -.000 056 -.000 040
Samples -.000 890 -.000415 -.000 417 - .002 634 -.000 253 -.000 310 -.000 098 -.000 215 -.000 035
.000 734 -.000 092 -.000 340 -.000 367 -.000 288 .000 008 -.000 616 -.000 117 -.000 182
200 Asymptotic -.000 200 -.000 097 -.000 070 -.000 550 -.000 266 -.000 191 -.000 058 -.000 028 -.000 020
Samples -.000 831 -.000 019 -.000 057 -.000 553 -.000 331 -.000 177 .000 096 -.000 095 .000 023
.000 140 -.000 064 -.000 069 -.000 919 -.000 223 -.000 200 -.000 135 -.000 210 -.000 086
e
....0'1N
e e
7.6.13. Estimates of covaria.nce of (P12(1.0), P2j{1.0)} from convergent samples in sets of 50 for r equal
3, 5, 7, by sample size
e
'12' P21 '12' P23 P12' PZ4
N----------': 3 s 7 3 S 7 3 S 7
S Asymptotic 0 0 0 0 0 0 0 0 0
Samples .002 177 .007 074 .004 235 0 -.000 526 .000 962 .000 798 .000 011 .000 798
.002 382 .001 652 .001 158 .000 168 .001 843 -.000 929 0 .000 876 .002 974
10 Asymptotic 0 0 0 0 0 0 0 0 0
Samples .006 825 .003 026 .000 469 .000 015 .000 760 -.000 179 .000 330 .000 916 -.001 392
.003 792 .003 226 -.000 622 .000 735 .000 053 .000 507 .000 993 -.000 260 -.000 513
20 Asymptotic 0 0 0 0 0 0 0 0 0
Samples .000 233 .002 399 -.000384 -.000 312 -.000 309 .000 062 .000 226 -.000 913 .000 041
.000 411 -.000 560 .000 602 .000 244 .000323 .000 592 .000 131 -.000 081 .000 726
50 Asymptotic 0 0 0 0 0 0 0 0 0
Samples -.000 577 .001 211 -.000 334 .000 430 -.000 322 -.000 026 .000320 -.000 205 .000017
.000 457 .000 545 .000 080 .000 138 -.000 023 .000 155 .000 024 -.000 609 .000 026
100 Asymptotic 0 0 0 0 0 0 0 0 0
Samples -.000 339 .000 377 .000 212 .000 023 -.000 155 .000 009 .000 139 -.000 017 -.000 038
.000 148 -.000 302 -.000 367 .000 088 .000 007 .000 081 -.000 041 .000 076 -.000 045
200 Asymptotic 0 0 0 0 0 0 0 0 0
Samples .000 074 .000 039 .000 000 .000 001 -.000 002 .000 014 -.000 070 -.000 067 -.000 047
.000 121 .000 024 .000 081 -.000 043 .000 042 -.000 036 .000 086 -.000 047 -.000 019.....0W
e e
7.6.14. Estimates of covariance of (P13(1.0), P2j(1.0)} from convergent samples in sets of 50 for r equal
3, 5, 7, by sample size
'",.
'" '" '",.
P13' P21 P13' P23 P13' P24
N-----------':' 3 5 7 3 5 7 3 5 7
5 Asymptotic 0 0 0 0 0 0 0 0 0
Samples .003 866 .002 586 .000 869 0 -.002 371 -.000 147 -.002 315 .003 387 -.001 749
.010 426 -.002 532 .000430 -.000 534 -.000 224 .000 825 0 .003 364 -.002 425
10 Asymptotic 0 0 0 0 0 0 0 0 0
Samples -.003 455 -.003 411 -.000 169 -.000 628 -.000 079 .000 399 .001 522 .000 225 .000 405
-.007 834 -.000 169 .000 759 .000 476 .000 386 .000 071 -.000 310 -.000 652 -.000 134
20 Asymptotic 0 0 0 0 0 0 0 0 0
Samples -.000 901 -.001 510 -.000 293 -.001 155 .000 398 -.000 011 .000 357 .001 515 -.000 476
.001 253 -.000 015 -.000 346 -.000 516 .000 629 .000 163 .000 077 .000 006 .000 027
50 Asymptotic 0 0 0 0 0 0 0 0 0
Samples -.000 362 -.000 435 .000 395 .000 284 .000 110 -.000 008 -.000 030 -.000 112 .000 013
-.000 238 -.000 328 -.000 239 .000 039 -.000 246 -.000 034 .000 089 .000 331 .000071
100 Asymptotic 0 0 0 0 0 0 0 0 0
Samples -.000 819 -.000 254 .000 011 -.000 101 .000 210 .000 005 -.000 038 -.000 141 .000 029
-.000 069 .000 066 -.000 051 -.000 053 .000 073 .000 045 -.000 174 .000 259 -.000 010
200 Asymptotic 0 0 0 0 0 0 0 0 0
Samples -.000 072 -.000 029 .000 011 -.000 046 -.000 029 -.000 036 -.000 035 .000 106 .000 045
-.000 034 -.000 028 -.000 049 .000 108 -.000 015 .000 007 .000 002 -.000 0~2 -.000 002
e
I-'0'~
e e
7.6.15. Estimates of covariance of (P14(1.0), P2j(1.0)} from convergent samples in sets of 50 for r equal
3, 5, 7, by sample size
P14' P21 P14' P23 P14' P24
N r 3 5 7 3 5 7 3 5 7
5 Asymptotic 0 0 0 0 0 0 0 0 0
Samples .001 039 -.001 918 -.000 531 0 .001 306 -.000 058 .001 116. .000 839 -.000 468
.003 998 -.000 726 -.001 447 .000 582 .001 580 -.001 140 0 -.000 092 .001 656
10 Asymptotic 0 0 0 0 0 0 0 0 0
Samples .000 071 .002 906 .000 679 -.001 133 -.000 663 -.000 640 .000 665 -.001 151 .000 529
-.000 434 -.001 619 -.000 743 .000 044 .000 201 -.000 161 .000 578 .000 542 -.000 693
20 Asymptotic 0 0 0 0 0 0 0 0 0
Samples -.000 465 .000 477 -.000 504 -.000 239 -.000 522 .000 170 -.001 701 -.000 498 -.000 231
-.000 214 .000 540 .000 106 -.000 386 .000 163 -.000 189 -.000 143 -.000 864 .000 859
50 Asymptotic 0 0 0 0 0 0 0 0 0
Samples .000450 -.000 406 .000 105 .000 232 -.000 061 -.000 045 -.000 218 .000 294 .000 213
.000 276 -.000 037 .000 212 .000 040 .000 058 -.000 035 -.000 037 -.000 052 .000 112
100 Asymptotic 0 0 0 0 0 0 0 0 0
Samples .000 020 -.000 142 .000 022 .000 073 -.000 054 -.000 038 -.000 185 .000 096 -.000 032
-.000 254 -.000 080 .000 007 -.000 030 -.000 008 .000 013 .000 104 -.000 002 .000 073
200 Asymptotic 0 0 0 0 0 0 0 0 0
Samples -.000 012 .000 091 .000 027 -.000 032 -.000 014 .000 015 .000 030 -.000 001 .000 021
.000 014 .000 117 .000 038 .000 038 -.000 042 -.000 004 -.000 015 .000 053 -.000 016
e
I-'
~
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7.6.16. Estimates of variance of modified (~lj) from convergent samples in sets of 50 for t' equal 1.0, r equal
3, 5, 7, by sample size
~12 ~13 t 14
~3 5 7 3 5 7 3 5 7
5 Asymptotic .029 230 .020 781 .017 593 .020430 .015 331 .013 200 .007 036 .005 126 .004 374
Samples .031 049 .062 051 .048 566 .044 142 .022 705 .037 739 .009 967 .003 186 .003 771
.073 266 .035 328 .035 971 .030 393 .027 064 .013 359 .012 611 .012 733 .018 043
10 Asymptotic .014 615 .010 390 .008 797 .010 215 .007 665 .006 600 .003 518 .002 563 .002 187
Samples .057 087 .026 609 .007 262 .011 199 .006 986 .004 416 .004 818 .006 012 .001 905
.018 669 .016 645 .008 069 .012 361 .007 614 .005 135 .003 748 .002 046 .001 814
20 Asymptotic .007 307 .005 195 .004 398 .005 107 .003 833 .003 300 .001 759 .001 282 .001 094
Samples .009 263 .011 405 .004 360 .006 259 .002 710 .003 019 .002 096 .001 957 .001 188
.012 819 .008 288 .006 989 .004 082 .004 188 .002 409 .001 312 .002 071 .001 617
50 Asymptotic .002 923 .002 078 .001 759 .002043 .001 533 .001 320 .000 704 .000 513 .000 437
Samples .003 495 .002 999 .001 776 .002 349 .001 484 .001 655 .000 851 .000 670 .000 494
.002 364 .003 508 .001 791 .003110 .001 413 .001 192 .000 698 .000 649 .000 516
100 Asymptotic .001 461 .001 039 .000 880 .001 021 .000 766 .000 660 .000 352 .000 256 .000 219
Samples .002 297 .001 283 .000 773 .001 002 .000 699 .000 671 .000 372 .000 269 .000 207
.001 393 .001 050 .000 793 .001 025 .000 643 .000 441 .000 458 .000 200 .000 181
200 Asymptotic .000 731 .000 520 .000 440 .000 511 .000 383 .000 330 .000 176 .000 128 .000 109
Samples .000 736 .000 555 .000 528 .000 404 .000 422 .000 296 .000 167 .000 126 .000 115
.000 603 .000 477 .000434 .000 613 .000 412 .000 328 .000 219 .000 116 .000 135
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7.6.17. Estimates of variance of modified (t2j ) from convergent samples in sets Qf50 for t' equal 1.0, r equal
3, 5, 7, by sample size
~21 ~23 ~24
~ 3 5 7 3 5 7 3 5 7
5 Asymptotic .722 890 .351 051 .252 641 .053 260 .026 094 .018 872 .130 554 .063 336 .045 555
Samples .225 866 .436 090 .192 892 0 .015 358 .000 689 .007 641 .041 973 .028 648
.347 900 .158 531 .151 468 .007 641 .004 326 .023 241 0 .061 200 .017 451
10 Asymptotic .361 445 .175 526 .126 320 .026 630 .013 047 .009 436 .065 277 .031 668 .022 777
Samples .655 673 .165 863 .122 311 .017 633 .006 081 .016 176 .045 302 .069 133 .032 361
.272 818 .225 843 .185 398 .023 883 .007 376 .001 585 .016 761 .015 256 .024 361
20 Asymptotic .180 722 .087 763 .063 160 .013 315 .006 524 .004 718 .032 638 .015 834 .011 389
Samples .202 308 .110 798 .108 904 .014 338 .007 235 .001 650 .036 840 .039 183 .026 376
.375 534 .172 301 .157 414 .014 566 .005 005 .008 864 .044 083 .033 137 .020 505
50 Asymptotic .072 289 .035 105 .025 264 .005 326 .002 609 .001 887 .013 055 .006 334 .004 555
Samples .076 271 .045 378 .044 639 .002 907 .001 409 .001 112 .012 821 .099 736 .004 977
.057 043 .043 200 .027 619 .002 542 .000 666 .000 597 .020 491 .018 083 .004 123
100 Asymptotic .036 144 .017 552 .012 632 .002 663 .001 305 .000 944 .006 528 .003 167 .002 278
Samples .053 481 .020 723 .013 419 .001 234 .000 417 .000 690 .009 214 .004 311 .002 695-)
.030 043 .021 054 .013 207 .000 562 .000 613 .000 291 .007 129 .002 994 .004 434
200 Asymptotic .018 072 .008 776 .006 316 .001 331 .000 652 .000 472 .003 264 .001 583 .001 139
Samples .023 336 .011 563 .007 175 .001 045 .000 326 .000 182 .002 774 .000 992 .000 674
.016 909 .007 122 .008 144 .000 263 .000 236 .000 185 .002 230 .001 555 .001 052
e
I-'0......
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7.6.18. Estimates of covariance of modified {tlj , t lk} from convergent samples in sets of 50 for
tequal 1.0, r equal 3, 5, 7, by sample size
t 12, t 13 t 12,t14 t 13,t14N_______r 3 5 7 3 5 7 3 5 7
5 Asymptotic .000 643 .000 437 .000 365 -.001 647 -.001 057 -.000 864 .000 019 . .000 013 .000 011
Samples .002 965 .015 427 .020 957 .001 071 .001 259 .000 812 .004 272 -.000 443 .006 184
.034 956 ~010 864 -.006 809 .018 945 .011 718 .019 682 .006 209 .001 290 -.002 855
10 Asymptotic .000 321 .000 219 .000 182 -.000 824 -.000 528 -.000 432 .000 009 .000 006 .000 005
Samples -.005 459 .001 992 .000 867 .002 836 .001 510 -.001 218 .001 127 .000 046 -.000 218
.004 673 .003 349 -.001 009 .001 094 .001 371 -.000 785 -.001 337 .000 827 -.000 005
20 Asymptotic .000 161 .000 109 .000 091 -.000 412 -.000 264 -.000 216 .000 005 .000 003 .000 003
Samples -.000634 .000 486 -.000 620 .000 627 .000 098 -.000 557 .000 187 -.000 599 .000 700
-.000 240 .002 385 -.000 234 -.000 797 -.000 265 -.000 035 -.000 048 .000 197 -.000 032
50 Asymptotic .000 064 .000 044 .000 036 -.000 165 -.000 106 -.000 086 .000 002 .000 001 .000 001
Samples -.000 123 -.000 222 .000 379 -.000 333 -.000 421 -.000 153 .000 424 .000 104 -.000 125
.000062 .000 034 -.000 201 .000 034 -.000 341 -.000 050 .000 037 -.000 173 -.000 055
100 Asymptotic .000 032 .000 022 .000 018 -.000 082 -.000 053 -.000 043 .000 001 .000 001 .000 001
Samples -.000 181 -.000 192 -.000 030 -.000 225 -.000 130 -.000 070 -.000 092 -.000 095 .000 001
-.000 083 .000 035 .000 002 -.000 113 -.000 124 .000 007 .000 060 .000 006 -.000 027
200 Asymptotic .000 016 .000 011 .000 009 -.000 041 -.000 026 .000 022 .000 001 0 0
Samples -.000 008 .000 055 .000 050 -.000 080 .000,032 -.000 048 -.000 024 -.000 012 -.000 029
-.000 029 .000 075 -.000 026 -.000 014 -.000 061 -.000 006 .000 015 -.000 017 .000 004
e
....0\00
e e
7.6.19. Estimates of covariance of modified (A2j , A2k} from convergent samples in sets of 50 for
t' equal. 1.0, r equal 3, 5, 7, by sample size
A21, A23
A21, A24 A23 , A24
N--------------.3 5 7 3 5 7 3. 5 7
5 Asymptotic .064 39.4 -.031 522 -.022 786 -.011 448 -.005 624 -.004 073 .000 502 .000 248 .000 181
Samples 0 -.012 657 -.000 845 -.007 463 -.OlD 580 .013 563 0 -.002 315 .000 200
-.012 291 .000 010 -.014 924 0 -.037 760 -.009 414 0 -.000 670 -.001 433
10 Asymptotic -.032 197 -.015 761 -.011 393 -.005 724 -.002 812 -.002 036 .000 251 .000 124 .000 090
Samples -.016 198 .003 531 -.002 396 .017 572 .023 898 -.012 060 -.002 762 .004 522 .006 412
.003 686 .001 245 -.003 410 .003 281 -.007 426 -.007 087 -.001 089 -.000 591 .001 372
20 Asymptotic -.016 098 -.007 880 -.005 696 -.002 862 -.001 406 -.001 018 .000 126 .000 062 .000 045
Samples -.005 724 .003377 .000 156 -.001 781 .004 159 .004 242 -.001 028 .003 269 -.000 168
-.009 658 -.003 548 .014 340 -.019 955 .009 581 .012 334 .001 349 .001 018 -.000 074
50 Asymptotic -.006 439 -.003 152 -.002 278 -.001 145 -.000 562 -.000 407 .000 050 .000 025 .000 018
Samples -.000 299 -.001 487 -.000 896 - .• 005 671 .001226 .003 628 -.000 548 .000 432 .000 224
-.001 170 -.001 485 -.000 561 -.002 011 .001 122 -.001 097 -.001 546 -.000 294 .000 154
100 Asymptotic -.003 220 -.001 576 -.001 139 -.000 572 -.000 281 -.000 204 .000 025 .000 012 .000 009
Samples -.001 680 -.000 639 -.001 233 -.004 184 .001 268 .000 155 .000 064 .000 014 .000 091
.000 996 -.000 235 -.000 391 .001 902 .001 065 .001 532 -.000 595 .000 290 -.000 189
200 Asymptotic -.001 610 -.000 788 -.000 570 -.000 286 -.000 141 -.000 102 .000 012 .000 006 .000 004
Samples -.001 901 -.000 458 -.000 250 -.000 626 -.000 636 -.000 204 .000 356 -.000 158 .000 015
. -.000 074 -.000 167 - .000 383 -.001422 -.000 172 -.000 208 -.000 021 -.000 139 -.000 027
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....()'\\0
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7.6.20. Estimates of covariance of modified {~12' ~2j} from convergent samples in sets of 50 for t equal
1.0, r equal 3, 5, 7, by sample size
~12'~21 ~12'~23 ~12'~24
~ 3 5 7 3 5 7 3 5 7
5 Asymptotic .057 415 .029 506 .021 885 -.001 218 -.000 668 -.000 511 .009 465 .004 658 .003 376
Samples .035 049 .111 462 .060 542 0 -.002 389 .002 098 .001 029 .003 417 .009 143
.092 952 .035 910 .020 807 -.001 349 .006 637 -.001 507 0 .003 486 .006 284
10 Asymptotic .028 707 .014 753 .010 942 -.000 609 -.000 334 -.000 255 .004 733 .002 329 .001 688
Samples .164 466 .040 274 .010 038 -.001 563 .005 440 .000 329 .015 155 .016 011 -.001 107
.041 758 .042 583 .017 252 .005 602 .000 263 .000 960 .003 804 - .001 238 -.000 697
20 Asymptotic .014 354 .007 376 .005 471 -.000 305 -.000 167 -.000 128 .002 366 .001 164 .000 844
Samples .013 238 .021 459 .003 605 -.000 165 .000 142 .000 266 .002 381 .002 254 .001 703
.044 172 .015 950 .021 501 .000 259 .000 607 .003 603 .001 511 .003 395 .005 024
50 Asymptotic .005 741 .002 950 .002 188 -.000 122 -.000 067 -.000 051 .000 946 .000 465 .000 338
Samples .002 653 .007 519 .002 113 .001 090 -.000 604 -.000 088 .001 329 .000 630 .000 632
.004 949 .005 043 .002 309 .000 163 - .000 325 .000 187 .000 929 -.000 179 .000 332
100 Asymptotic .002 871 .001 475 .001 094 -.000 061 -.000 033 -.000 026 .000 473 .000 233 .000 169
Samples .001 747 .002 855 .001 686 -.000 179 -.000 ?38 -.000 058 .000 645 .000 316 .000 123
.003 244 .000 665 -.000 044 .000 151 -.000 019 .000 067 .000 434 .000 506 .000 353
200 Asymptotic .001 435 .000 738 .000 547 -.000 030 -.000 017 -.000 013 .000 237 .000 116 .000 084
Samples .001 760 .000 899 .000 607 -.000 064 -.000 045 .000 037 .000 083 -.000 110 -.000 040
.001 664 .000 606 .000 906 -.000 040 .000 030 -.000 099 .000 283 .000 024 -.000 001
e
~o
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7.6.21. Estimates of covariance of modified {A13 , ~2j} from convergent samples in sets of 50 for t'
equal 1.0, r equal 3, 5, 7, by sample size
t13
, t21
t13
, t23
t13
, t24
N~ 3 5 7 3 5 7 3 5 7
5 Asymptotic .010 961 .006 651 .005 318 -.008 478 -.005 367 -.004 363 -.000 102 -.000 061 -.000 049
Samples .048 177 .019 208 .026 571 0 -.004 504 -.000 444 -.004 056 .007 263 -.003 721
.056 685 .005 514 .004 042 -.002 902 -.000 530 -.000 613 0 .008 490 -.003 107
10 Asymptotic .005 480 .003 325 .002 659 -.004 239 -.002 684 -.002 182 -.000 051 -.000 031 -.000 024
Samples .015 157 -.005 822 .002 521 -.002 998 -.001 395 -.000 927 .002 311 -.002 149 -.000 603
-.021 700 .008 679 .002 941 -.001 743 -.000 157 -.000 151 .000 087 -.001 458 -.000 369
20 Asymptotic .002 740 .001 663 .001 330 -.002 119 -.001 342 -.001 091 -.000 026 -.000 015 -.000 012
Samples -.000 194 -.002 119 -.000 701 -.003 324 -.000 150 -.000 263 -.000 152 .002 181 -.001 206
.007 778 .006 246 -.000 318 -.002 525 -.000 102 -.000 964 .000 421 .000 336 .000 837
50 Asymptotic .001 096 .000 665 .000 532 -.000 848 -.000 537 -.000 436 -.000 010 -.000 006 -.000 005
Samples -.000 274 -.000 414 -.000 451 .000 067 -.000 197 -.000 335 .000 098 -.000 372 .000 019
.000 813 -.000 043 -.000 111 -.000 138 -.000 366 -.000 III .000 206 .000 833 .000 172
100 Asymptotic .000 548 .000 332 .000 266 -.000 424 -.000 268 -.000 218 -.000 005 -.000 003 -.000 002
Samples -.001 686 -.000 334 .000 366 -.000 189 .000 111 -.000 101 -.000 133 -.000 280 .000 075
.000 028 .000 631 .000 047 -.000 150 -.000097 -.000 030 -.000 295 .000 582 .000 024
200 Asymptotic .000 274 .000 166 .000 133 -.000 212 -.000 134 -.000 109 -.000 002 -.000 002 -.000 001
Samples .000 248 .000 224 .000 152 -.000 196 -.000 138 -.000114 -.000 114 .000 209 .000 059
.000 270 .000 171 -.000 039 -.000 013 -.000 094 -.000 052 -.000 015 .000 022 -.000 012
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f-i""-JI-'
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7.6.22. Estimates of covariance of modified (~14' ~2j) from convergent samples in sets of 50 for r equal
1.0, r equal 3, 5, 7, by sample size
~14' ~21 ~14'~23 ~14'~24
~ 3 5 7 3 5 7 3 5 7
5 Asymptotic .001 729 .001 184 .000 990 -.000 103 -.000 062 -.000 049 -.010 749 -.005 597 -.004 179
Samples .010 266 -.006 212 .003 059 0 .002 174 -.000 107 .000 266 -.000 657 -.003 122
.032 401 .006 920 .000 482 .000 532 .003 918 -.002 653 0 -.004 372 -.000 897
10 Asymptotic .000 865 .000 592 .000 495 -.000 051 -.000 031 -.000 024 -.005 375 -.002 798 -.002 089
Samples .010 896 .010 765 .002 489 -.001 887 -.001 157 -.001 304 -.002 632 -.005 981 -.001 014
-.000 034 .000 904 -.002 859 .000 371 .000 558 -.000 323 -.000 879 -.000 358 -.002 920
20 Asymptotic .000432 .000 296 .000 248 -.000 026 -.000 015 -.000 012 -.002 687 -.001 399 -.001 045
Samples -.000 512 .001 563 -.000 737 -.000 428 . -.000 733 .000 353 -.004 813 -.003 531 -.000 364
-.000 870 .000 086 .001 669 -.000 510 .000 048 -.000 104 -.002 808 -.003 531 -.000 364
50 Asymptotic .000 173 .000 118 .000 099 -.000 010 -.000 006 -.000 005 -.001 075 -.000 560 -.000 418
Samples .001 724 -.001 182 .000 480 .000 408 -.000 023 -.000 011 -.001 247 -.000 367 -.000 087
.001 164 .000 198 .000 805 .000 154 .000 102 -.000 015 -.001 427 -.000 814 -.000 245
100 Asymptotic .000 086 .000 059 .000 050 -.000 005 -.000 003 -.000 002 -.000 537 -.000 280 -.000 209
Samples .000 146 -.000 536 .000120 .000 102 -.000 026 -.000 045 -.000 985 -.000 264 -.000 243
-.000 653 -.000 127 -.000 061 .000 067 .000 042 .000 052 -.000 477 -.000 228 -.000 215
200 Asymptotic .000 043 .000 030 .000 025 -.000 002 -.000 002 -.000 001 -.000 269 -.000 140 -.000 104
Samples .000 061 .000 273 .000 060 -.000 082 -.000 012 .000 017 -.000 133 -.000 071 -.000 036
.000 178 .000302 .000 140 .000 045 -.000 026 -.000 017 -.000 242 -.000 064 -.000 129
e
t-'.....N
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:,
e'
7.6.23. Estimates of variance of (~i.(0.2)) from convergent samples,J "
in sets of 50 for r equal 7, by samp1, size
N '12 '13 '14 '21 '23 '24
10 . A$ymptotic .000 496 .000 587 .000 172 .017 257 .000 350 .004 191
Samples .000 217 .000 489 .000 146 .043 048 0 .031 689
.000 399 .000 597 .000 287 .036 857 0 .006 874
'"
SO Asymptotic .000 099 .000 117 .000 034 .003 451 .000 070 .000 838
Samples .000 094 .000 137 .000 034 .003 878 .000 021 .000 963
.000 077 .000 169 .000 032 .004 412 .000 121 .001 146
200 Asymptotic .000 025 .000. 029 .000009 .000 863 .000 018 .000 210
Samples .000 019 .000 032 .000 010 .000 839 .000 021 .000 172
.000 024 .000 026 .000 077 .001 218 .000 024 .000 191
e
....'-IW
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7.6.24. Estimates of covariance of{~.. (0.2)~'k(0.t)}from'convergent samples1J 1 "
insets of 50 for t equal 7, by sample size
e
N i>12,Pl3 1>12,1>14 1>13,1>14 1>21'1>23 1>21'1>24 1>23,1>24
10 . Asymptotic -.000 018 -.000 005 -.000 006 -.000 030 -.000 365 -.000 007
Samples -.000 048 -.000 Q31 -.000 002 0 -.005 761 0
.000 026 -.000 052 -.000 018 0 -.002 164 0
50 Asymptotic -.000 004 -.000 001 -.000001 -.000 006 -.000 073 -.000 001
Samples -.000 003 -.000 006 -.000 000 -.000 031 ...000 337 -.000 016
.000 003 -.000 005 -.000 008 .000 053 .000 169 .000 005
200 Asymptotic -.000 001 -.000 000 -.000 000 -.000 001 -.000 018 -.000 000
Samples -.000 001 -.000 001 .000 002 -0000 015 0000 057 -.000 015
-.000 001 -.000 002 -.000 002 .000 030 .000 024 -.000 013
I-'.......j:l-
175
7.6.25~ Estimates of covariance of fi>lJ(0.2),i>2k("o.2)} from c9nvergent
sa~p1es in sets'of 50 for r equal 7, by sample size
.N '12"21 '12,1523 1>12"24 ~13'~21 t>13'~23
10 Asymptotic 0 0 0 0 0
Samples - .000017 0 -.000 318 .000 118 0
-.000 195 0 -.000 073 -.000 079 0
50 Asymptotic 0 0 0 0 0
Samples -.000 093 -.000 001 .000 103 .000 090 .000 006
.000 090 -.000 008 .000 041 -.000 260 -.000 011
200 Asymptotic 0 0 0 0 0
Samples .000 016 .000 003 .000 006 -.000 034 .000 000
.000 009 .000 005 .000 001 .000 021 .000 002
N ~l3'~24 ~i4'~21' 1514,1523 1514,1524 ,,-
10 Asymptotic 0 0 0 0
Samples .002 309 -.000 061 0 .000 394
.000 892 .000 068 0 .000 383
50 Asymptotic 0 0 0 0
Samples .000 020 .000 095 .000 001 -.000 037
-.000 081 -.000 052 .000 014 .000 019
200 Asymptotic 0 0 0 0
Samples .000 008 .000 022 . -.000 002 -.000 005
-.000 008 -.000 009 -.000 002 -.000 008--
e e
7.6.26. Estimates' of variance of modified' l1.ij
) from convergent samp1~s in sets o·f 50
for t' equal 0.'2, requa1 7, 'b-y sample size
N t 12 ~13 t 14 ~21 k23 ~24
:
10 Asymptotic .015 665 .016 450 .004 739 .573 380 .010 039 .119 '705
Samples .007 Z81 .007 531 .003963 1.069 757 0 .347 461
.012 829 .017 413 .007 615 .674 638 0 .321 655
50 Asymptotic .003 133 .003 289 .000 948 .114 676 .002 008 .023 941
Samples .003 065 .003 862 .000 980 .144 691 ' /.000 512 .027 905';
.()02 567 .004 674 .000 832 .168 366 .001 132 .036,870
200 Asymptotic .000 783 .000 822 .000 237 .028 669 .000 502 .005 985
Samples .000 663 .000 893 .000 293 .028 344 .000 476 .004 952
.000 773 .000 757 .000 214 .000 388 .000 50S .005 358
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7.6.27. Estimates of covariance of modified (1ij
.;1ik
) from convergenf samples in sets
of 50 for ~ equal 0.2, r equal 7, by sample size
e
N I t 12,t13 I t 12,t14 I t 13,t14 I t 21,t23t
21,t
24t
23,t
24
10 Asymptotic .• 000 014 I -.000 178 I. .000 000 I - .010 415 -.002 679 .000 017
Samples -.000 649 .001 072 I - .001 318 I 0 I -.086 584 I 0
.002 093 -.001 056 -.000 370 0 - .067 447 I 0
50 Asymptotic .000 003 -.000 036 .000 000 -.002 083 -.000 536 I .000 003
Samples .000 148 -.000 189 .000 032 -.000 899 I -.007 566 I -.000 419
.000 091 -.000 209 -~OOO 145 .002 467 .008162 .000 168
200 Asymptotic .000 001 -.000 009 .000 000 -.000 521 -.000 134 .000 001
Samples ,000 000 -.000 037 .000 064 1-.000 377 I .002 172 I -,000 381
-.000 028 -.000 072 ' -.000 069 I .000 779 I .001 017 I -.000 315
.......,
.....
178
7.6.28. Estimates of covariance of modified (11j ,12k) from convergent
I' .samples in sets of 50 for t equal 0 •.2, r equal 7, by sample size
N t 12,t21 A12,A23 A12,A24 A13
,A21 t 13 ,A23
10 Asymptotic .009 699 -.000 012 .001 869 .000 994 -.000 929
Samples .017 765 0 -.004015 .000 605 0
.019 823 0 -.000 867 .014 903 0
50 Asymptotic .001 940 -.000 002 .000 374 .000 199 -.000 186
Samples .000 074 -.000 026 .003 390 .003 3'04 .000 148
.005 326 -.000 202 .001 749 -.008 141 -.000 346
200 Asymptotic .000 485 -.000 000 .000 093 .000046 - .000 046
Samples .001 021 .000 065 .000 236 -.001.030 -.000 028
v 000 742 .000 118 .000 060 .000 660 .. 000 039,
N A13
,A24 A14,l21 t14
,t23 l14,t24
10 Asymptotic .000 017 .000 243 -.000 002 -.002 094
samples -.001 918 -.018 310 0 -.003 830
.033 611 -.013 518 0 .010 929
50 Asymptotic .000 003 .000 048 -.000 000 -.000 419
Samples .000 451 .003 054 .000 042 -.001 558
-.002 895 -.001 924 .000 421 -.000 102
200 Asymptotic .000 001 .000 071 -.000 000 -.000 105
Samples .000 188 .000 565 -.000 048 -.000 206
-.000 186 - .000 388 -.000 032 -.000 309
e..
e
7.6.29. Estimates of. variance of {~ij(2.0)) from conver'gent sal!1ples in
sets of SO for r equal 7, by sample size
N ~12 ~13 ~14 ~2l ~23 ~24
10 Asymptotic .004 776 .008 144 .003 465 .051 ,376 .018 541 .027 833
Samples .005 579 .007 645 .002 260 .050 987 .015 686 .046 340
~006 555 .005 941 .003 475 .035 S76 .021 642 .025 643,,
SO Asymptotic .000 955 .001 629 .000 693 .070 275 .003 708 .005 566
Samples .000 586 .,001 971 , .000 696 .009 SSO .004 468. .005 367
.000 614 .000 978 .000 663 .012 594 .005 061 .005 232
200 Asymptotic .000 239 .OQO 407 .000 173 .002 569 .000 927 .00,1 392
Samples .000 266 .000 551 .000 232 .002 510 .000 887 .001 574
.000 281 .000 706 .000 137 .002 620 .000 936 .001 716
e
to-'
~
e e
7.6.30. 'Estimates of. covariance of r'ij(2.0)"ik(2.<1)} from convergent
samples in sets of 50 for r equal 7, by samp1/ia size
e
N I '12'~13 '12"14 I '13''14 1~21'~23 I '2P~24 I '23'24
10 Asymptotic -.001 424 -.000 493 -.000 988 -.007 780 ~ -.012 454 1 -.003 074
Samples -.000 268 .000 083 -.002 126 -.016 209 -.032 774 .001 873
-.002 814 - .001 356 .000 940 -.007 623 -.022 612 -.006 399
SO Asymptotic -.000 285 -.000 098 -.000 198 -.001 556 -.002 491 1 -.000 615
S.amp1es - .000 383 -..000 095 -.000 149 -.002 1991 -.002 7321 -.000 368
-.000 172 - .000 253 -.000 124 -.0020391 -.001 581 I -.001 531
200 Asymptotic -.000 071 -.000··025 -.000 049 ~.ooo 389 -.000 623 -.000 154
Samples -.000 109 -.000 032 -.000 055 -.000 35>3 -.000 822 -.000 211
-.000 175 -.000 048 -.000 008 -.000 603 -.000 746 I .000 148
....000
181
7.6.31: Estimates of covariance of (Plj(2.0),PZk(2.0») fro~ convergent
samples in sets of 50 for r equal 7, by sample size
N ~12'~Zl ~12'~23 ~12'~Z4 '13"21 '13"23
10 Asymptotic 0 0 0 0 0
Samples -.003 917 .002 334 .003 493 -.005 046 .001 977
-.000 436 .000 132 .001 429 .000 265 -.000 622
50 Asymptotic 0 0 0 0 0
Samples -.000 189 .000 372 .000 659 .000 763 -.000 671
-.000 107 .000 0.28 .000348 .000 331 -.000 442
200 Asymptotic 0 0 0 0 0
Samples -.000 084 .000 110 .000 057 .000 000 .000 015
-.000 010 .000 065 -.000 080 -.000 023 -.000 028
N ~13'~24 '14'~21 ~14"23 ~14'~24
10 Asymptotic 0 0 0 0
Samples .002 535 -.021 543 -.001 685 -.003 193
-.001 55Z· -.002 361 -.001 334 .001 443
50 Asymptotic 0 0 0 0
Samples -.001 014 -.000 181 -.000 109 -.000 235
.000 232 -.000 281 .000 044 .000 101
200 Asymptotic 0 0 0 0
Samples -.000 077 -.000 036 .000 088 -.000 010
-.000 108 .000 085 -.000 015 .000 055
e
7.6.32.
..,e
Estimates of variance of modified (t .. ). from conver.gent samples in sets. . . 1.J
of 50 for t' equal 2.0, requal 7, by sample. si~e
e
N I t 12"" i:14
.:.... t 21~ t 23 I 1:24l13
10 Asymptotic .014 772 .006 513 .002 482 .190 048 .020 216 I .024 590
Samples .021 878 .005 9.51 I .002 079 I .166 406 I .009 164 I .040 766
.021 292 .007917 .002 5.50 .069 658 .012 555 .•013 940
50 Asymptotic .002 9S4 .001 303 .000 496 .038 010 .004 043 .004 918
Samples .002 426 .002 028 .000 583 .040307 .002 944 .005 581
.004 743 .000 847 .000 577 .057 938 .. 002· 556 I .007 533
200 Asymptotic .000 738 .000 326 .000 124 I .009 502 .001 011 I .001 229
Samples .000 726 .000 470 .000 192 I .010 956 I .000 246 I .001 483
.001 078 I .000 577 I .000 1301 .012 672 I .000 578 I .001 993
....coN
e -(
e
7.6.33 • 'Estimates of covariance of ,modified ('tij,'tik
) from coiivergEmt samples
in sets of SO fort' equal 2.0, 'requal 7, by sample size
e
N I t 12,t13 t 12,l14 t13
,t14 t 21,t23t 21,t24 t
23,t24
10 Asymptotic .001 069 -.001 095 .000 029 -.030 238 -.002 7'89 .000 470
Samples -.013 337 .001 166 .000 363 -.020 588 I -.035 360 I .004 843
.001 135 -.000 680 .002 171 -.004 434 -.020 738 -.002 371
SO Asymptotic .000 214 -.000 219 .000 006 -.006 048 -.000 S58 .000 094
Samples -.000 008 -.000 500 .000 142 -.003 233 -.001 063 .000 867
.000 059 -.000 80S -.000 000 -.001 544 .007 532 -.000498
200 Asymptotic .000 053 I -.000 055 I .000 001 I -.001 512 I -.000 139 ~OOOO23
Samples .000 059 -.000 070 .000 005 -.000 476 -.000 314 .000 026
-.000 092 -.000 113 .000 072 .000 009 .000 867 .000 439
....cow
.,.
184
7.6.34. Estimates of covariance of modified (A1j,A2k) from convergent
samples in sets of 50 for r equa12.0, r equal 7, by sample sUe
.N t 12,t
f1 ~\·2Jt23 t 12,i:24 l13t 21 t13Jl23"". .
,10 Asymptotic .029 966 -.002 174. .003 396 .008 282 -.005 341
Samples .021 878 .000 299 .006 564 .009 175 -.001 082
.013 606 -.000 059 .000 214 .004 954 -.002 983
50 Asymptotic .005 993 -.000 435 .000 679 .001 656 -.001 068
Samples .004 913 ..000 130 .001 739 .003 194 -.001 219
.012 128 .000 152 .003 704 .001 332 -.000 719
200 Asymptotic .001 498 -.000 109 .000 170 .000 414 -.000 267
Samples .001 514 -.000 053 .000 176 .000 579 -.000 060
.002 196 .000 333 .000 401 .000 361 .000 130
N l13,t24 t 14,t21 t14Jt23 t14Jl24
10 Asymptotic -.000 326 .000 691 -.000 114 -.004 030
Samples -.000 876 .Oll 100 -.001 563 .013 448
-.001 451 .001 061 -.000 482 -.001 594,
50 Asymptotic -.000 065 .000 138 -.000 023 -.000 806
Samples -.000 872 -.000 320 -.000 217 -.001 122
.000 465 -.000 926 .000 165 -.001 293
200 Asymptotic -.000 016 .000 034 -.000 006 -.000 202
Samples -.000 070 .000 091 .000 050 .000 303
-.000 185 .000 047 .000 024 -.000 255
!
e e « e
" 'A
7.6.3S. Estimates of mean square error of (Plj(1.0)} from convergent samples, ,
in sets of 50 for r equal 3, 5,7, by sample si~e
... .. I ...P12 P13 P14
~ 3 5 7 3 5 7 I 3 I 5 I 7
5 .014 142 .010 266 .OOS 365 .016310 .009 056 •• 008 719 .004 630 I .002 500 I .001 712
.010 219 .004 967 .005 720 .011 429 .012 640 .007 466 .004 681 .004 028 .005 542
10 I .002 936 .003 441 .002 440 .005 434 .003 502 .002 209 .002 411 .003 234 .001 246
.003 808 .002 695 .,002 095 .006 982 .003 845 .003 016 .002 899 .001 294 I .001 138
20 I .003 104 I .002 521 I .001 402 I .003 288 .001 683 .00l 583 .001 064 .001 198 .000 663
.001 70S .001 747 .001 075 .002 128 .002 117 .001 371 .000 879 .001 063 .001 091"
SO I .000 841 .000 772 .000 538 ,.001 286 .000 871 .000 819 '.000 494 I .000 415 I .000 336
.000 665 I .000 954 .000 497 .001 722 .000 778 .000 700 .000 401 .000 372 I .000 321
100 I -- 7121 __ 350 .000 203 .000620 .000 432 .000 388 .000 184 .000 161 I .000 121
..000 278 .000 580 .000 381 ' .000 278 .000 288 .000 128 .000 115.000 369.000 343
200 I .000 202 1.000 172 .000 154 .000 227 .000 231 .000 161 .000 100 .000 078 .000 071 I-'00
.000 123 I .000388 I .000 248 I .000 194 I .. 000 108 I .000 080 I .000 069Vt
.000 157 1.000 145 I
e e
7.6.36. Estimates of mean square error of (P2.(1.0)} from convergent samples in, J
sets of SO for r equal 3, 5, 7, by sample size
e
~5
, 10
20
50
100
200
'21 '23 I '24
3 I 5 I 7 3 5 I 7 I .3 I 5 I 7
.057 188 1.053 564 .039 109 .000 968 .008 289 .001 248 .010 032 I .019 740 I .011 942
.058 021.040 170 .034 134 .005 305 .003 655 .012 481 .. 007 350 .032 610 I .010 406i '
.079 7431.031 995 I .026 799 .009 418 .002 993 .008 010 .014 554 .023 087 I .013 305
.061 070 1.035 024 .023 299 .009 997 .004 670 .001 577 .•010 968 .009 594 .010 758
.046 511 .020 847 .021 374 .008 780 .004 146 .001 160 .016 205 .016 942 .008 053
.045 320 .019 167 .020 953 .008 145 .003 799 .004 734 .019 331 .012 214 .008 053
.014 901 .010 274 .008 124 .002 560 .001 356 .001 077 .007 272 .004 549 .002 335
.016 236 .009 575 .006 817 .002 314 .000 952 ..000 791 .010 160 .005877 .002 074
.013 214 .004 374 .003 356 .001 290 .000 611 .000 648 .004 50S .002 048 .001 328
.007 032 1.004 605 .003 151 .001 048 .• 000 730 .000 455 .003 580 .001 .553 .002 042
.006 185 1.002 634 .. 001 710 .000 983 .000 305 .000 214 .001 488 .000 509 .000 367I-'
.004314 I .001 702 .001 996 .000 472 .000 309 :'000 176 .001 341 .000 849 .000 523,' 000'\
~~~--~
e e -( e
7.6.37. Estimates of mean product error of (Plj(1.0),Plk(1.0») from convergent samples
in sets of 50 for r equa1,3, 5, 7, by sample size
P12'~13 '12"14 '13"14
~ 3 5 7 3 I 5 I 7 3 I 5 I 7
5 -.004 114 .•000 552 . .001 696 -.000 932 -.001 140 -.001 084 .001 021 -.000 6971 .000 588
I -.002 386 .001 786 -.002 914 .000 748 .000 748 .003 912 -.000 603 -.000 931 ~ -.001 143
10 1-.000 771 .000 136 .000 153 .000 141 .000 364 -.000 688 -.000 421 -.000 976 I .000 342
-.002 094 .•000 156 -.000 77S .000 S23 .000 III -.000 036·1 -.001 291 I .005 176 I -.000 275
20
50
100
200
-.000 7931-0000 ?21 -.000 392 .000 392 .000 086 -.000 192 -.000 138 -~OOO 600 I .• 000 302
-.000 578 .000 270 -.000 203 -.000 112 -.000 022 -.000 062 -.000 167 -.000 063 I -.000 146
-.000 1751-.000 245 I .000 069 -.000 173 -.000 109 -.000 081 .000 194 I .000 013 I -.000 116_.
-.000 126 I -.000 066-.00017.4 -~OOO 106 -.000 140 .000 048 -.000 133 -.000 039 -.000 116
-.000 147 -.000 141 -.000 067 -.000 051 -.000 018 -.000 022 .... 000 078 .... 000 054 1 -.000 018
-.000 086 .... 000 084 -.000 032 -.000 033 -.000 027 ....000 008 -.000 113 -.000 063 I -.000 047
-.000 027 .... 000 014 -.000 001 -.000 027 .000 022 -.000 024 -.000 039 -.000 021 -.000 028....
-.000 0611 .000 009 I -.000 042 I .... 000 003 I -.000 042 I -.000 004 I -.000 005 .... 000 036 .... 000 013 00.....:
e e
7.6.38. Estimates of mean product error of (P2j(1.0),P2k(1.0)} from convergent samples
in sets of 50 for r equal 3, 5, 7, by sample size
e
~5
10
20
50
100
200
..... '" '21"23 '21"24 '23"24
3 I 5 7 3 5 7 - 3 5 7
.003 141 -.003 464 .001 313 .002 196 -.000 928 .000 578 .004 046 -.004 876 -.000 965
-.004 699 -.000 342 ,-.006 611 .001 369 :'.001 103 -.001 338 -.000 049 -.019 834 I -.004 970
-.004 155 -.000 216 .002 426 I-.001 022 -.000 871 .000 272 -.003 231 -.000 9231-.007 984
-.000 581 .000 381 -.001 913 -.000 842 -.000 376 .000 616 !002 114 -.004 337 -.003725
-.005 107 -.000 400 -.001 245 -.001 739 -.000 316 -.000 696 -.'004 486 -.004 450 -.002 532
-.004 999 -.001 173 .001 619 .000 505 .000 158 -.000 411 -.007340 -.003 997 .000 332
-.000 780 I -.000 632 -.000 684 -.000 526 -.000 037 .000 046 -.003 671 1-.000 784 1 .000 040
.000 040 I -.000 664 -.000 216 -.001 436 -.000,288 .000 026 -.002 927 -.000 951 I -.000 919
-.001 101 I -.000 415 -.000 406 -.000 068 -.000 214 -.000 039 -.002 634 -.000 246 -.000 333
.000 661 -.000 092 -.000 333 -.000 640 -.000 117 -.000 222 -.000 397 -.000 303 -.000 005
-.000 931 -.000 009 -.000 057 '.000 100 - .. 000 082 .000 023 -.000 571 -.000 321 -.000 188 ,....000 149 I -.000 06S I .... 000 041 I -.000 113" -.000 210 I -.000 082 I -.000 891 I -.000 217 -.000 185 00
00
e e
7.6.39. Estimates of mean product error of (P12(1~O),P2j(1.0») from convergent
samples in sets of 50 for r equal 3, 5, 7, by sample size
e
~12'~21 ~12'~23 I ~12'~24
N 3 5 7 3 I .5 I 7 I 3 I 5 I 7
5 t ....005 046 .005 136 .000 888 -.002 227 I -.001 160 I -.000 211 I -.004 252 I -.000 659 I .000 084
I.002 422 -.000 153 -.000 349 -.000 963 .001 431 -.000 771 -.006 074 .001 838 .002 826
10 I .006 612 .002 523 .000 344 -.000 164 .000 452 .000 051 .000 051 .001 470 '.-.001 544
.002 801 .003 085 -.000 434 .000 825 .000 014 .000 257. -.000 223 -.000 845 -.000 450
20 i' ,.000 018 .002 151 -.001 296 -.000 251 -.000 289 .000 011 .•000 217 -.000 518 .000 084
.000 271 -.000 520 .000 562 .000 332 .. 000 439 .000 634 .000 151 .000071 .000 737
50 I -.000 574 .001 220 -.000 335 .000 429 -.000 325 -.000026 .000 322 -~OOO 211 .000 019
I.000 570 .000 545 .000 050 .000 177 -.000 023 .000 160 - .. 000 022 -.000 609 .000 012
100 I -.000 410 .000 376 .000 225 .. 000 047 -.000 155 .000 011 .•000 149 - .. 000 020 -.000 042
.000 130 I -.000 294 -.000 370 .000 103 I .000 007 I .000 072 I -.000 035 I .000 058 I -.000 1126
200 I .000 053 I .. 000 042 I .000 001 I .000 006 I .000 002 I .. 000 014 1-.. 000 069 - .. 000 063 ....000 048
•000 113 I -.000 019 I .. 000 097 I -.000 050 I .000 042 I .000 032 I .000 066 I -.000 044 -.000 017 ....~
e e t e
N
.5
7.6.40. Estimates of mean })roduct error of (P13 (LO),P2j (LO)) .fro~ convergent
samples in sets of SO for r equal 3, 5, 7, by sample size
'13"21
I'13"23 I '13"24
3 I 5 I 7 3 I " 5·1 7 3 I 5 "I 7
.004 354 I •.002 973 I .000 257 I .000 151 -.002 2451-.000 361 1_-.001 974 .003 521 I -.001 879
.010 395 I -.005 018 I .001 143 I -.000 671 -.000 791 .000 750 .004 633 .004 689 I -.002 355
10 I -.007 425 I -.OO@ 177 I -.000 161 I -.000 595 -.000 065 .000 384 .001 573 -.000 033 .000 415
-.003 325 .000 069 .000 617 .000 439 .000 467 .000 2.59 .000 191 -.000 412 -.000 182
20 I -.000 561 -.001 455 -.000 103 -.001 251 .000 394 -.000 122 .000371 .001 443 I ....000 383
.001 269 /-.000 009 I -.000 267 -.000 526 .000 647 .000 079 .000 075 .000 029 .000 004..
50 I -.000 353 -.000 425 -.000 394 .000 282 .• 000 107 -.000 008 -.000 023 -.000 119 .000 011
-.000 403 -.000 263 .... 000 323 -.000 018 -.000 264 -.000 020 -.000 156 / .000 340 I .000" 033
100 I -.000 675 I -.000253 .000 01S -.000 150 .000 210 .000 006 -.000 059\-.000 138 .000 028
-.000 870l .000 1158 -.000 055 ....000 060 .000 073 .000 033 -.QOO 021 .000 277 .000 015
200 I -.000 173 "". -.000·'029 .000 015 -.000 022 -.000 030 -.000 036 -.000 031 .000 lOS .000 042
-.000 042 I '\".000 ;045 -.000 049 .000 102 -.000 014 .000 007" -.000 018 -.000 032 -.000 002....1.00
e e t' e
7.6.41. Estimates of mean product error of (P14(1.0),P2j(I.0)} from convergent
samples in sets of,50 for, r equal 3, 5, 7, by sample size
I~14"21 I ·~14'~23 I '14"24
~~ 3 5 7 3 I .5 I 7 I 3 I 5 I 7
5 .O()O 741 -.001 230 .000 362 .000 216 .001 531 .000 255 I .001 607 I .001 077 ~ -.000 278, ,
I .004 002 -.000 238 -.001 940 .000 480 .001 691 -.001 088 -.000 558 -.000 3521 .001 607
10 1-.000 487 -.002 567 .000 680 -.001 274 -.000 871 -.000 643 '.000 440 -.000 778 .000 531
-.000 778 I -.001 357 -.000 659 .000075 .000 258 -.000 272 .000 156 .000 607 -.000 665
20 1-.000 485 I .000 392 -.000 S02 -.000 233 -.000 515 .000 169 I -.001 702 -.000 387 -.000 230
-.000 148 I ,.000 532 I .000 055 I -.000 428 .000 139 -.000 135 -.000 152 -.000 895 .000 874
50 I .000 462 -.000 410 .000 100 .000 229 -.000 060 -.000 043 -.000 210 .000 297 .000 224
.000 405 -.000 029 .000 216 .000 085 .000 056 -.000 036 -.000 089 -.000 051 .000 114
100 I .000 078 -.000 143 .000 036 .000 053 -.000 054 -.000 036 -.000 193 .000 094 ~.OOO 037
'"'.000 215 -.000 072 .000 009 -.000 062 -.000 008 .000 020 .000 091 -.000 021 .000 086
200 1-.000 018 .000 090 .000 025 -.000 031 -.000 015 .000 015 .000 030 -.000 002 .000 020....
.000 018 I .000 116 I .000 038 I .000 041 I -.000 042 I -.000 004 I -.000 000 .000 053 -.000 016 \01-"
e
7.6.42.
e
Est~ates of mean square error of modified (tlj ) from convergent samples in sets
of 50 for i equal 1.0, r equal 3)/ 5, 7, by sample size
e
~5
10
20
50
100
200
/
t 12 1:13 1:14
3 5 7 3 5 7 3 t 5 I 7
.044 868 .077 402 .061 676 .044 909 .022 958 .039 746 .010 008 I .003 396 I .003 808
.101 861 I .041 840 .042 102 .031 379 .032 062 .013 515 .013 680 I .012 763 I .018 531
.063 312 I .032 085 .008 162 .011 368 .006 99,S .004 416 .005 263 I .006 445 I .001 9.13
.009 232 .012 383 .007 629 .005' 190 .~·991 I .002 050 I .001 871.021 563 .016 808
.009 488 .012 501 .004 677 .006 323 .002 710 .003 350 .002' 105 .001 958 I .001 188
.013 961 .009 598 .007 482 .004 082 .004 224 .002 490 .001 336 .002 097 .001 635
.003 564 .003 012 .001 794 .002 378 .001 484 .001 655 .000 882 .000 671 .000 498
.002 393 .003 612 .001 794 .003 174 .001 446 .001 228 .000 734 .000 649 I .000 516
.002 298 .001 286 .000 783 .~l 090 .000 702 .000 671 .000 381 .000 271 .000 218
.001 394 .001 059 .000 818 .001 059 .000 6.54 .000 457 .000 483 .000 214 .000 190
.000 737 .000 569 .000 569 .000 438 . .000 422 .000 297 ~OOO 167 .000 127 .000 116....
.000 616 .000 491 .000 463 .000 640 .000 412 .000 328 .000 220 .000 116 .000 135 \0N
e e.( • e
7.6.43. Estimates of mean square error of modified (~2j} from convergent samples in sets of
50 for r equal 1.0, r equal 3, 5, 7, by sample size
t 21t 23
t 24
~ 3 5 7 3 5 7 3 5 7
5 .233 557 .442 538 .193 542 0 .015 851 .000 735 .014 267 .041 973 .028 656
.373 340 .160 932 .152 034 .007 987 .004 636 .025 574 .010 000 .066 002 .017 452
10 .675 385 .170 596 .123 680 .018 288 .006 657 .020 120 .045 319 .074 728 .032 373
.273 489 .227 624 .193 734 .026 842 .008 205 .001 784 .019 819 .016 488 .024 913
20 .202 323 .111 190 .117 112 .017 564 .008 819 .001 967 .037 298 .044 196 .027 362
.385 001 .117 543 .161 421 .017 624 .006 605 .011 304 .044 918 .034 826 .020 998
50 .077 776 .045 378 .045 031 .003 748 .002 054 .001 536 .012 938 .010 018 .005 038
.058 548 .049 284 .029 942 .002 911 .000 814 .000 816 .021 332 .018 302 .004 287
100 .058 325 .001 286 .013 544 .001 453 .000 702 .000 854 .009 214 .000 271 .002 716
.031 016 .001 059 .013 250 .000 747 .000 654 .000 362 .007 129 .000 214 .004 665
200 .024 054 .000 569 .007 292 .001 565 .000 422 .000 249 .002 780 .000 127 .000 682
.016 910 .000 491 .008 916 .000 359 .000 412 .000 256 .002 351 .000 116 .001 065
....\0to)
194
7.6.44. Estimates of mean ~quare error of (Pij(0.2») from convergent
samples in sets of 50 for r equal 7, by sample size
N ~l2 ~l3 ~l4 ~21 ~23 ~24
10 .000 229 .000 490 .000 157 .044 846 .000 002 .032 456
.000 428 .000 599 .000 288 .037 110 .000 002 .006 876
50 .000 096 .000 137 .000 034 .003 923 .000 022 .000 983
.000 077 .000 170 .000 032 .004 738 .000 121 .001 147
200 .000 019 .000 032 .000 010 .000 840 .000 021 .000 190
.000 024 .000 026 .000 007 .001 218 .000 025 .000 191
7.6.45. Estimates of mean square error of (Pij
(2.0)} from convergent
samples in sets of 50 for r equal 7, by sample size
N P12 1513"
P14 P2i ~23 ~24
10 .006 199 .008 340 .002 262 .053 508 .015 690 .046 661
.006 910 .006 007 .003 590 .041 177 .021 178 .025 647
50 .000 587 .001 997 .000 710 .009 673 .004 471 .005.566
.000 614 .000 982 .000 681 .012 662 .005 100 .005 514
200 .000 271 .000 583 .000 238 .002 546 .000 887 .001 641
.000 284 .000 717 .000 140 .002 625 .000 964 .001 803
7.6.46. Estimates, of mean square error of modified (tij ) from
convergent samples in sets of 50 for ~ equal 0.2,
'r equal 7, by sample size
195
N t t13
t14
t21
t23
t 2412
10 .007 886 .007 602 ' .004 273 1.117 543 0 .349 327
.014 147 .017 564 .007 638 .675 071 0 .322 811
50 .003 669 .003 883 .000 988 .145 306 .000 522 .028 700
.002 603 .004 693 .000 839 .183 201 .001 291 .036 879
200 .000 663 .000 896 .000 294 .028 478 .000 543 .005 567
.000 775' .000 763 .000 214 .041 097 .000 622 .005 361~
7.6.47. Estimates of mean square error of modified {t .. J from1.J
convergent samples in sets of 50 for t' equal 2.0, r
equal 7, by sample size
N t 12 t 13t 14 t 21 t 23 t
24
10 .026 962 .007 372 .002 109 .166 782 .012 026 .042 746
.022 813 .007 917 .002 701 .075 742 .017 413 .013 951
50 , .002 452 .002 068 .000 618 .040 446 .003 646 .005 989
.005 104 .000 848 .000 577 .059 778 .003' 554 .008 214
200 .000 726 .000 497 .000 194 .010 979 .000 358 .001 544
.001 112 .000 587 .000 137 .013 434 .000 722 .002 162
196
7.7 Multivariate Tests
7.7.1 The;Hypothesis that a Mean Vector and a Covariance Matrix Are
Equal to a Given Vector and Matrix
Given the k-component observation vectors Yl'Y2""'YW from
N(~,Q), the likelihood ratio criterion for testing the hypothesis H:
~ =~o' Q := Qo' is~ W -1
e = (i)2 IBQ~112 e-%[trBQ~l + W(y - ~o)'Qo (y - ~o)] ,
where B =I (Ya - y)'(Ya - y).a
When the null hypothesis is true, -2 log e is asymptotically distributede
as x2 with ~k(k + 1) + k degrees of freedom.
In the application of the test in section 4.2.1 to the {PiJ(t')},
say, the observation vector is
Thus k equals 6, and the degrees of freedom are 27.
7.7.2 The Hypothesis tha~ Two Sets of Three Variates Each Are Independent
Given 6-component observati~nvecto~s Y1'YZ" •• ,YW' from N(~,Q),
the likelihood ratio criterion for testing the hypothesis HI that the
sets of variates {Ya l'Ya2'Y(3) and {Ya4'Ya5 'Y( 6) are independent is
:I --l!L.IB r·1 BI '1 2
where B := [bij
] , i,j = 1,2, ••• ,6,
B := [b . .J, i,j = 1,2,3,1 1J
B = [bij
] , i,j =4,5,6,2
197
and bij =I (Yai - Yi )(Yaj - Yj)' . The exact formula for the size of
athe region is
= B (W-4 1)->z 2' 2
1"' (W_2)r'Yl-3)+ ~ 2, z(W-6)/2 (2/l-z(-!- _ -!-)
6I;r:W-6)r(W;4) W-6 W-3;
_ ...L ,z?5.(7( _ 2 . -1.;;,) + iL 1 (l~))W-5 S1n z W-4 oge Ii '
where B is the Incomplete Beta Function and r the GamSiFunction. Ifz
the number ofobservatioti vectors is large, W(l-z) is ~pproximately
2distributed as X with 9 degrees of freedom.
In the application of the test in section 4.2.2 the two sets of
e-
•
....
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INSTITUTE OF STATISTICS(Mimeo Series available for distribution at cost)
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