ORTHOGONAL STATISTICS INVOLVING THETHIRD AND FOURTH SAMPLE MOMENTS

FOR NEGATIVE BINOMIAL DISTRIBUTION

byPeter Shih—Shiang Hsing

Thesis Submitted to the Graduate Faculty of the

Virginia Polytechnic Institute

in candidacy for the degree of

MASTER OF SCIENCE

in

Statistics

August, 196hBlacksburg, Virginia

6

Q ..

TABLE OF CONTENTS

Chapter Page

I I I I I I I I I I I I I I I I Lp

(a) Orthogonal Polynomials.... „ . . . . b(b) Orthogonal Statistics ......... 6

II THE NEGATIVE BINOMIAL DISTRIBUTION • • • • • 9

III ORTHOGONAL STATISTICS FOR PARAMETERS OF THENEGATIVE BINOMIAL DISTRIBUTION • • • • • „ „ lb

(a) General Method of Obtaining OrthogonalPolynomials • • • • • • • • • • • • ••(b)

Relationships Between Q's and Sample

Moments for Negative Binomial Distri—

I I I I I I I I I I I I I I II(0)

The Use of Recurrence Relationships „ • 17(d) Tables of Expected Values of

I I I I I I I I I I I I I II(e)

An Example of the Use of the Tables of

EXPGCÜGÖ.IV

O • • • • • • • • • • • • • • • • •

I I I I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I I I I I I I I I I

... ..

TABLE OF CONTENTS (coht•)

Chapter Page

I I I I I I I I I I I I I I I I I I I I I

IAPPENDIXB (l) •....„........ • . . . . . A2

I I I I I I I I I I I I I I I I I I I I

I IQ

I I I I I I I I I I I I I I I. I I I I I

N

CHAPTER l

INTRODUCTION

The purpose of this thesis is to extend the development

of tables of orthogonal statistics for the negative binomial

distribution. Shenton and Myers Enl have introduced the

concept of orthogonal statistics for securing the first few

moments of moment.estimators and used this development to

investigate small sample properties of moment estimators

involving the first two sample moments. The purpose of this

work is to extend this development to treat statistics

involving the first four sample moments.

(a) Orthogonal Polynomials

Let F(x) be a distribution function with finite moments

of all order. Then there exists a set of polynomials qr(x)

such that the following orthogonality condition hold:

oo

JqI_(x)qS(x)dF(x) = ÖI, for r ’* S

-oo

* O for r f s

where Ör > O, ÖO * l, qO(x) = l and the coefficient of xr

in qT(x) is unity. The polynomials will be infinite in

number if the set of points of increase of F is infinite,

„ 5 -

and finite in number if F has only a finite number of points

of increase (Cramer [1+]). These orthogonal polynomialsare well known for many of the classical distributions. For

example, the first few polynomials for the standard normal

distribution (often called the Hermite polynomials (Cramer[1+] are:

Hom) = 1X

T": xa "• 1

7d3(x) x - 3x

etc.

The orthogonal polynomials for the case of a Poissonvariate with parameter m can be found from the generatingequation:

where A is an advancing difference operator (i.e.,Ax(r) = rx(r”l)) and x(r) is the factorial termx(x-l)(x-2)....(x-r+l). These polynomials are called theCharlier's polynomials (Szego [l3])§ the first few ofthese are:

qo : l

ql x x — m

)

q2 “ x2 - x - 2xm + mz

q3 T x(x—l)(x-2)—3mx(x—l)+3xm2·m3

Certain recurrence relationships exist from which these

polynomials can be generated for other distributions, The

g most general expression for say, the rth polynomial qr hx) can

be written asH

= r (r) —li„(r) r—2„ I (r)„„ (r)x x*c O (l_l)where c(§} T coefficient of xl‘in qY(x)

Other useful relationships, particularly for products

of the q's and methods for finding the polynomials in general

are given in Chapter III,

(b) Orthogonal Statistics

Let (xl,x2,..,,xH) be a sample of independent observation

from a distribution for which the orthogonal polynomials qT(x)

exist, The rth orthogonal statistic (Shenton and Myers [ll])

is defined as H

H • 1-*2H < >/ < >

As a result of the orthogonality property of the q's, we

can writeE(Qr) ¤ O (r=l,2,,..)

(1-3)

- 7 -

where 6r,S * O for r f s

= l for r = s

It will be noted from (l~l) and from the basic definitionof the q's that Q? can be written as a linear function ofml, m2„„„„mT, where mk is the kth crude sample moment i•e•,

7 n3*1

3Thus of course mi can be written as a linear function

of Ql;Q2,„•.QT• (The expression relating the m's and Q's ingeneral and for the specific case of the negative binomialdistribution will be given in Chapter III)•

If we desire to obtain the first few moments of amoment estimator t(mimémé„..mi) and if we expand this statistic(or say the pth power of the statistic if we desire to obtain

the pth moment) in the form

1 1 a11o...oQ1Q2+&1o1o...oQ1Q3+ *•••¤ B Ei

then the corresponding crude moment can be obtained by takingexpectations term by term of the powers and products of the

}

Q*s, and these expected Q-products can be expanded in ascending

T„

3powersof l/n with the coefficients being the expected values

of products of the q*s. Associated with these orthogonal

statistics are the following tables which can be found in

Shenton and Myers [ll];

(i) Expected Q-products in terms of expected q-products

to order ten through terms in n°5 (in general).

(ii) Expected Q-products to order eight through terms

inn”h

for the negative binomial distribution in terms of

the pcpulation parameters for products involving Ql and Q2.

Because of the difficulties involved in obtaining the

maximum likelihood estimators for negative binomial parameters

(Bowman [j3]), much attention has been drawn to the investi-

gation of moment estimators.

Tables are presented in this thesis which are an

extension of those described by (ii) above toproductsinvolving

Q1, Q2. Q3 and Qu. These tables will then aidinthe

determination of the bias, variance, etc. of

momentestimatorsinvolving the first four sample moments.

T

T

- 9 - 2 V

cameras .2 VEgg ggQgEggg glNOMIAL gggEglBUTION

The negative binomial distribution is one of the mostwidely used of the two parameter discrete distributions,having applications in biological data, health and accidentstatistics, psychology, and other fields. A literaturereview may be found in Bartko [ 2].

There is a similarity of the negative binomialdistribution to the positive binomial distribution. Theprobability generating function for the letter is (pt + q)n.

VIf we replace p by —p and n by —n, we arrive at the generating Vfunction of negative binomial distribution with mean np and V

variance np(1+p). Note that the variance exceeds the mean

Vfor the negative binomial distribution.There are various parametric forms for the negative

V

binomial distribution. we shall consider primarily one form, V

namely that due to Anscombe [J.] with prcbability function:V

x 0,1,2,...V x a > 0 , k.> 0 1

with parameters A and a. (Here the combinatorial part istaken to be unity when x = 0).

VThe mean and variance for this particular form are k

and N + K2/d respectively. Important and useful analytic

- 1Q -

properties include the cumulant generating function

-dln[l—(d+k)(eit-l)/A] and the factorial moment generatingfunction (l-kt/d)—°. Other parametric forms include thatdue to Fisher ['7] with parameters s M A/d and k = d, more

recently a form given by Evans [ 5] with parameters a * K/d

(same as p in the Fisher form) and m r k, and the Pascalform [12] with probability function

XFF—l T X x * 0,1,2,....X >

P q , r an integer, p+q=l

There are various models which give rise to the nega—

tive binomial distribution. For example Feller [ 6]

describes what he calls "Apparent Contagion" i.e., the

compounding of the Poisson distribution. If the Poisson

parameter m varies and is considered to have a statistical

distribution described by a Gamma frequency function givenbv

X Bk k—l -BmeB > 0, m > O, k > O.

then the frequency generating function becomes Ioo

J em(t"l)£(m)dm [l+(l-t)/B]"k <B+t>1)

>o

The resulting probability functicn isPrax = x> = ——‘——-—)—·—-—$-"*+1 k—"ä°—*@=—’=l <—I§.—[;>k<~j_—};—[;>Xb

B - ll—If

we set k = d and ß * d/R we arrive at Anscombe'sform of the negative binomial distribution.

The Poisson Distribution is regarded as a limiting formof the negative bincmial distribution. In the Pascal form,if we let pel, q+O and N be fixed such that q * N/r, thenthe moment generating function of the negative binomialdistribution, pr/(l—qe@)r, becomes the moment generating

0function of the Poisson Distribution, ek]- -1), when r+oo„The negative binomial probability function, then, becomesthat of the Poisson with parameter R.

Other known distributional forms result [ 2] under

certain physical models and limiting conditions.

Estimation Qi Parameterg

Ne shall limit ourselves to the problems of estimationof the parameters for the form of Anscombe• The maximumlikelihood estimators are given by the equations,

* eR mlH +••l-4}-A -{-IOO+v 1 na 2 na

*= ...,..,.,.............,..,....,.,, -...,..„.._„.,,,..,,.,,.,,,..,.....„ -{- • • •}} (1,* vwl

+ —-ä ,O d +R—l

where ni is the observed frequency of xi, R is the maximumR

value of i such that i * O,l,2,..„R. N * E Hi •1*0

u

— l2-Thereare iterative methods of solving (2-l). Bowman

[3 ] has investigated some sampling properties of the

maximum likelihood estimator of u (Actually this work wasnot restricted to the Anscombe form; the maximum likelihood

estimators for the form of Fisher and Evans were alsoconsidered).

Due to the difficulty in solving the likelihoodequation, the method of moments is usually used for theestimation of parameters. Equating the first two moments

yields the following estimators

^ .K = ml

Fisher Ü'?] gives a range of the parameters for whichthe asymptotic efficiency exceeds 90%. Shenton and Myers

[ll] investigated the small sample properties of E (firstfour moments through terms in

n°“and covariance determi-

nant through terms inn_3)

and showed that in the range ofthe parameter space in which the asymptotic efficiency ishigh, the expansion for the variance becomes "explosive",

i.e., higher order terms are large in comparison to then”l

terms, even when we consider terms through n°h. This

suggests that even in the range of high asymptotic

efficiency, the merit of Ä is questionable and it might be

instructive to consider the investigation of estimators

I- 13 -

. involving moments of higher order, say m3 or mh.

Other estimators have been suggested e.g., use of mean_ and proportion of zeros, inverse moment, and geometric

moment. An estimator combining the method of moments and

that of using the proportion of zeros has also been

mentioned [5 J particularly for the parametric form ofEvans.

... lg} ..

CHAPTER III

ORTHOGONAL STATISTICS FOR PARAMETERS QF THENEGATIVE QLNOMIAL Qlßßäläßglgß

(a) General Method_gf Obtaining Qgthogonal Polynomials

Let f(x) be a frequency or probability functien with

E(x) E u and all moments existing. Let X = x - u. The

q's are obtained in the following manner (see Kendall and

Stuart_

) 1 XQ (X Z ‘ •l 1 ul

1 X X2

q2(x) Ä l Ül Q

where uj = jth central moment for the distribution of x

Thus ql(x) = X , (3-1)

q2(x) E X2 * XU3/U2 “ H2 • (3*2)

(1

- 15 - 1In general, the rth orthogonal polynomial qr(x) can be

written in determinantal form as follows:

qP(x)=(—1f‘1 X X2 ... XT 1 ßl ßz ... nP_l

1. I Q I Q •Q“1

Q2 “3•·· uril5ur—1

gr “r+l°"“2r-1

Based on (3-1), (3-2) and the basic definition of Q's

given by (1-2), we arrive at

Q1 “ mi ' Q ·Q2

m2Conversely,then we can write

mi :2 + Ü 3

m2 = u2 + Ql<2u + u3/u2) + Q2 ·

(b) Relationships Between Q*s and Sample Moments jp; the

Negative Binomial Distribution

Shenton and Wellington [12] give an inverse moment

relationship for the case of the parametric form of Anscombe;

— lo — ‘xh) (1 "‘ ?~·/(1-:1)°lQ'I‘”l‘1I-(za) , (

1 where x(T) * x(x—l) ••• (x-r+l)„ (Here A is a symbolicdiffererential operator, i•e„, Aqr(x) * rqr_l(x)).

ee can use the form above and the definition of the

Q*s to express the sample faetorial moments in terms of theQ*s„ For example

x(l) = (l+k/aA)°ql(x) T ql(x)+kAql(x)+f(O) ,

Summing both sides over the sample (xl,x2,...xH),

H (1) H · (2 x. /11 =·—· E L<1l(x.)+?»]/¤1j::lJ 3::1JH(1)

Q1 H H ‘

Using the same procedure we can obtain:

*

II1(2) Q2+2>\.(1+1/<1)<;ll‘**K‘°‘(l‘*‘l/OL)

1Herem(k) refers to the Kth factorial sample moment,

H (i•e•, jil xj(xj—l)•••(xj-k+l)/n •

Conversely, we can write

Q1

(~ 17 —

I

,_ rv » .. ’(3/-

i3(1+2/a)(1+1/a) .

(C) Ehe Has Qi äenarrensaHelatiaaabinsSzego[13] gives a general fundamental recurrence

relationship for the q's in the form

q;w* (x-ar)qT_l-brqT_2 , (3-3)

where r E 2, qo = 1 and ql * x · aO in this case• (Note theargument x has been omitted). Myers [ 9] give the relation-

ships among ar, br for the parameters of negative binomial

distribution as followsz

arbr

* R(N+d)(r~l)(d+r-2)/@2 • (3-5)

In developing the expected powers and products oftheQ'sin terms of the parameters A and d, it is useful to have1

expressions involving crdinary products and powers of theI

q*s. For example we have the expressions developed by IMyers[9]qlqr_l

” qr+qr_l(r—l)(d+2K)/d+qr_2(r—l)(d+r-2)A(k+d)/ag,((3-6.)I(

_ (11in__11.11__.....__....._..................................................J

) (( - ie -(

q2qT_l * qr+l+q;2(r—l)(e+2h)/d+qT_l(r—l)?2(d+r-l)k(H+d)

+qr_§2(r-l)(r—2)(d+r—2)k(N+d)(di2K)/ag

+qP_¥(r~l)(r—2)(¤+r-3)(arr-3)ä2(%+dF/¤u • (3-7)

All q—products can be written as linear functions of

individual q*s„ Another useful expression developed by

Shenton [lO] is

E(q§) - tr = NT(N+¤)P(¤+r—l))”)r!/ag? , (2-8)(1*) „,+, .!.„ ·’) ..2.,*where (q+r-l) (ü T~l)(ü·P-¤)•··(d·l)d •

From (3—8) it can be verified that

Öl Z »$2 = 2N2(k+d)£(a+l)/ag ,

From (3-Ö), (3-7) and (3-Ü) we can derive certainexpected q—products involving ql and q2• For example if we

need, say E(qäq2) we can set r*3 in (3~o), multiply by ql

take expectations, the result is,very simply

ZNotehow we have taken advantage of the orthogonality

property of q'S•

- 19 -

At this point let us insert an explanation of notation

to be used throughout the remainder of the text, We shall, „ „ üyp r

~‘, d;Q r §· ~ „· d Q r E·; Pefsr to ¤(Ql J2 @3 Q2) aß (l es

[1“ßP3Thä] (The same notation will be used in the Appendices)•Since our goal is to extend the tables of orthogonal

g statistics to involve Q3 and Qu, it is important that we

develop recursion formulae similar to (3-0) and (3-7) for

If we set r S 3 in (3-A) and then express ar in terms ofag, we have

ar S a3 + (r-3)(2%+d)/d • (3-9)

From (3-3), for r S 3 we can write

and thus(x—a3) * q3/q2+b3ql/qg • (3-10)

If we substitute (3-9) into (3-3) and insert theexpression in (3-10) for x - a3, we obtain

we then transfer the product q3qT_l to the left-hand side (1and obtain l

(3-12) contains q-products which then must be expressed as

... QQ ..

11ueer ruuetfeus er individual q's„ We eeu use <3—5)<3-6)and (3-7) to expand qgqh, qlqF_l, q2qP_l and brq2qr_2•After much simplification the result becomes

+ qr_j3(r—l)(r—2)(r—3)(¤+r-3)(¤+r-2)%2X2n¢/a5+ qr_h<r—1><r—2>tr-3)<e+r-A>(e+r-3)<e+r-2)w3x3/e?, „

(3-13) pwhere 2% + a ~ n and % + d = X . J A

Using a similar procedure, we obtain for qh;

+qr+i2(r—l)[2(a+r+l)%X+3(r—2)n2]/az+

qP_3¤2<r-1)<r-2><r-2}(e+r—2><e+r-2>A2x2E2<e+r-1)XX

+ qr_uxA<r—1><r—2)<r-3)(r-A><e+r—4><e+r-3}<e+r-2>x2X2n/J7

I— 21 —

i qT_5(r-l)(r—2)(r-3)(r—h)(¤+r—5)(u+r—u)(¤ir-3)(¤+r-2)

As an illustration of the use of (3-13) and (3~lL),

suppose we need to obtain, say [123h].

[123h] Z EIqhqßKqgiqzän/e+ql2(d+l)NX/¤2]}Z Eqh[q3q3+q3q22n/d+q3ql2(d+l)kX/dg]

+ qh 12nß/a2...+qh 2(a+1)kX/a2+...] ,I

(Note that the terms not shown in the expression above willI

result in zero contribution after multiplying by qh and

Itaking expectation). I

Z @h[9akX+36KX+l8n2+l2n2+2dkX+2KX]/agI

Z 2A(a+1)(¤+2)(a+3)k?Xh(11ahX+38AX+3On2)/ag „I

As another example, suppose we wish to find the term[ii;]. I3 Z 2,[lÄ J E(qhQlqh)

If we let r Z ? in (3-1L) and evaluate qlqh, we have: I

E I‘ E Cxhiqhqojqä hn/¤¤"‘¤1LLq3 Z»(<1+3)NX/@2}

11..1111Z..11Z1Z..1.........................._..._._______________________J

— 22 - Lwe can use the recursion formulae in a similar manner toevaluate qhqö, qä, and qgqh, the result becomes

2 [11,3] Eql+{ . . .qh2l;,O[?tX1'}(cx+‘L,,) ( 3«m<+15;¤.x+2n2) 3/aß • ‘

...+qh96n[3%X(a+4)(dkX+5kX+4n2)+n+]/a5'°

...+qh96kXn(d+3)[3(e+4)%X+n2]/d5+....}

’* 1152 (@+1) (@+2 ) (@+3 )?~·l‘Xl"¤ { 57~·X(@+l+) ( 3@?\·X+157~·X

+ znz >+2[3m<(¤+z.) <¤w»><+-6>»><-isz.,¤¤·2>+·n’*J+ 2>(x<e·¤—;> <2m«.x·#-12>»><+·¤··°=> l/am

(d) Qgble gf Expected Values gi Q-products

As was mentioned in Chapter I, the expectation of

products of the Q°s can be expressed in terms of expansions

in powers of l/n with the coefficients being the [l°2ß3rh§]

terms. In fact, in general the expansion (see Myers [*9])

will be . b „ .’

ksa

where 8

g(Here{Z} denotes the integral part of Z). (rasBtP...)Ä,which represents the coefficient of n-2 in the expansion isexpressed in terms of the expected q—products. Shenton and

Myers [ll] give tables for the (rüsßtruä) in terms of the

„ 23 „

(J ' im . .Erasptyuä] for any fourQ’s

in powers not exceeding ten

through terms in n—5. These were used as an aid in. d„ß r i . _ „constructing tables of (l A 3 A ) in terms of A and d, for

a + B i r i ggf 3, through terms in nun. These tables areshown in Appendiees A, B and C. An explanation of the useof the tables is given at the beginning of the appendiees.Of course (3-T3), (3-7), (3-13) and (3-lu) were used toobtain the required

[l“2b3TA€]„As a brief illustration of

how these tables were constructed, suppose we consider theexpansion of E(Q%). From the tables of Shenton and Myers[ll], we have the expression for (3h) as

LP 2 3 ‘ _ 6 , 2 E(3 ) 3[3*l /¤L· ··LL {(.3*] -· A32] }/n -_ ’)

From the recurrence formulae, L3h] and [3“]werederived,resulting in the final expressions,

_„ 7 „.· e‘

lO108aiegm,

')~-. ' 2 ·· ·. n '”*» q -( ··» ·x ·— ·\ Ö(eemkwnt):

- -, ) 'D {J , · wg l·1· A3 J/(xl ,

where a = d + 1, b = d + 2, c E d 3 3, d = d + A, e — a + 5,f e d + s, g · a + 7, n = AA + a, and X L X + a.

- gn -

<e) de Exanala ei ehe Ess ei the.Tahlee ef Eeaaeäaa Q;.änadaateUsing the tables in the Appendices, one can investigate

the expansions of properties such as bias, variance and

higher order moments, etc., for moment estimators of the

negative hinomial parameters invelving the first four

sample moments. These expansione can be investigated through

terms inn”b.

Suppose we consider the moment estimators of

d and L found by equating the first moment and the ratio of

the 3rd to the 2nd factorial moment, i.e.,A I/\• E j

m . u 3 + Oand 1 1 1-+-%@1 mi + em; ,m(2)

“(2) *2(l+l/¤)

The estimator of d becomes^ ‘> T JÜ.A

Suppose we consider, say the bias of d. The sample

faetorial moments are easily expressed in terms of the Q'sA

and thus we can expand a asA n R g n , In3·

Q · . ,...6

+ QlQ2[e3+2d2—dk(d+3)]/2Aß(a+l)

OCC! O

- 25 - 1

Note that for the purpose of illustration we are onlyexpanding through powers cf 2 in order to find the biasthrough terms in n”l. In practice, of course, we wouldwant to go to higher order terms.

For example if we desired to express the bias to terms· in

n—2,we would expand d through terms of order A in the

' Q*s.

Upon taking expectation of d in (3~l5), we have,

E(d) * d + ä i2d(l2)l/K—(22)ld(d+3)2/§ß(d+l)ä

— (32)ld/?hé(d+l®+(l2)l(d3+2d2—®dE—l8k)/@A3(dilü

-· (13 )l 3 ((7.+3 )/E’¤2(¤c"‘“lX- ( ÄÖÜ ) lüg (C/.+3 ) (@+1)/’Ö§€A·5 (<1+lE}

+ ...,

we can substitute expression for (3£)l, (l2)l, (l3)l and(23)l found in Appendix A. Expressions for (l2)l and (22)l

are, of course, found in the tables of Shenton and Myers.The result becomes

E(d) cx [ZX-2X2(d·=—3)2/d2>\.(d·i¢i·l)-3X2(d·«2)/dL'7‘„3]+...,

A similar procedure could be performed for powers of Äif we desired to find higher order moments.

- 26

-CHAPTER IV ISUMMARY

This work is an extension of the development oforthogonal statistics in estimation problems for the negative Ibinomial distribution. The method can enable one to investi- Igate the sampling properties of moment estimators involvingthe first four samplemoments.The

stimulation for this work resulted out of curiosity )

concerning the properties of moment estimatorsinvolvinghighorder moments. If we call qr(x), (r B 0,1,2,...) the

Iset of orthogonal polynomials associated with the negative Ibinomial distribution, then the rth orthogonal statistic, )

defined for the sample xl,x2,...xn is II]QI,(x)

=‘jiil qI,(xj)/n

IExpectations of powers and products of the Q's can be

Iexpressed as expansions in powers of 1/n with the coefficients

Ibeing the population parameters. Myers and Shenton [11], who Iused this method to investigate sampling properties of moment Iestimators of the negative binomial parameters involving the Ifirst two moments, constructed tables of expectations

ofpowersand products of the Q's involving Q1 and Q2 throughÄ

order 8, to terms inn”h.

Appendices A, B and C of this Ithesis give expansions involving Ql, Q2, Q3 and Qu. Assuming I

I

- 27 -

i an estimator, say t (ml, m2, m3, mh) (or powers of t for thecase of variance and higher order moments of t) can beexpanded as a linear function of powers and products of theQ‘s,

one can take expectation term by term of this expansionand, through the tables in Appendices A, B and C, obtain anexpansion of the desired moment of t through terms in n°h.

This technique can be used to investigate such properties asbias, variance and higher order moments.

Tables of E(QäQ§Q§Qä) obtained in [ll] in terms of theexpectation of products of q's were helpful in this work.However there still remained the problem of going fromE(qäqg...) to the population parameters. Certain recurrence

formulae (See Chapter III) involving products of the q'swere developed to use in this task. Expressions for q3qT_land qAqr_l, r * 1,2,..., as linear functions of theindividual q*s were derived. Then by manipulating theseformulae and their products, for appropriate value of r,

the necessary expected q-products were found.

I I-23- II

IIIIII

I

I

I- 29 — IAPPENDIX

Tables of Lxpected Q-products involving Ql, Q2, Q3 andQu through terms in

n—h.

.3¤·EL><...h.;;l..@.-r1e.‘;.ie1.¤addAppendixA contains cases involving Q1, Q2 and Q3.

Appendix B(l) contains expected Q's involving dl, Q2, Q3 andQh, but only through order 5. Appendix B(2) contains thebalance of those involving the first four Q's, i.e., order6, 7, and 8 but rather than being expressed in terms of thepopulation parameters, they are expressed as functions of thesquare bracket terms (the expected q—products). Appendix C

contains expressions for the expected q-products that arenecessary for the complete use of Appendix B2, i„e., squarebracket term necessary in evaluating expected Q—products oforder 6, 7 or 8 involving Ql, Q2, Q3 and Qu. As an exampleof the use of Appendices B2 and C, suppose we desire theexpansion of E(Q2Q3Q2). For the scope of this work, thisIinvolves only one terms, namely the coefficient of

nӀ,i.e.,

From Appendix Btz) we see that (234h)h*l5[42]2[234].From Appendix C we note that [42] 2häbCÄ#Xh/@7 and[234] * l44abchhXun/dg. By substitution then we can easilySee that

I

„ BQ -

s£I.l".1;°.äNE2.l.2<§ A

* Expected Q-Products Involving gl, gz and Q3

AI.<;1¤.A1>.i.@.¤X + 6 “ X 2bXX + ng ~ jBk ¤ 6 · n QAX 1 ng k

6 = l I a ÖQAX + ng — pc;‘· E I; b <iRX ”* Zng I‘ q

6 * 3 * 6 2dXX + Bng 1 r6 # L · d ?dXX + nä 6

CL 6; 1. .}» »; 1. t

6 = « * f e;X # Eng * u

6 + 7 v6 * Ü ; h emX(f&X ¤ bug) + nh ” w

- ~i— V :: :.6 — J 1 f„„ — n y

QEAX + Qng · z

* Those yroducts involving only Ql and Q2 can be found in

tables by Shenton and Myers [lll

ÄP — Bl -

Products

0

Exgéusäs 9i.Q;Qs; 2(33)2 = Béabpnßxßn/ag

°‘ 3Ö8.bk7\•3X3/(17

(212)2 · 0

(322)2 · znabmßxßn/A6

(32l)2 s éabngxß/A5

Azxßssäs Qi Qxésx A

12a bnßxö/A8

(323)2 0

.. 32 ..

(331)}(332)}(323)}

pi .Q.¤d.6; 221606161296 aßbzknéxé/612

- 33 ..

(3322)3 “ 72abph5X5n /all

(32122)3 = l2abKhXh(akX+3k)/ag

(32l22)3(31’+)3

= ¤(3132)3

Böazbußxß/agzssagbnßxßn/ag

PP

+ 3ÖOcdeq+9Obcdk+3ÖOcdeq+l8abcqP

+ 20cdep+9Obcdk+9ab2k+l8abcq+9abgk+abp—9aabp) I

+ 2A2X2(l35Ocdpnb+25cdqu+27Ocdpn2+27Ocdqu+5hbckq (

+ 9abpn2+5hbckq+l8b2k)+9kX(2ucq2s+Öcpq2+Öcpq£+Abk2p) P3*P

I- ..11..111.1..11.......................................__.____________________1

.. ...

I+abc+3ab2~501„ab )+37»2 X2 (L+Ocdu

+ 9Ocdn2+l2bcq+Sbck+Lb2k

+ Babnz )+6 7~.X( Scqs+9cq2+2c;vq*+bkp‘*‘*3bk2 )+6 pgn ]/alz

(3h2)A = 24abx?x3[9£%x3(50Cd@£mX+200¤den2

+ zabzk-6¤abk)+27u2x2n2(loocdnß

+ Sb 2k+abp ) +5 L,7~„X( 2L;,qs'n2·+6 pqnz

(33l2)h 36abX?X3n[3OcdX?X2+XX(42cq+4cp+3bp+ l5bk+3ab—ap)+9pn2]/alO

(3322)A“ 72abX3X3n[2X?X3(6Ocde+3abc+2Ocd@

- 2uab)+3X?X2(6Ocdu+l2bck+lSOcdn2

+ 6abn2+lOcdj+ab+Sacq+2abk)

+ 6kX(l2qs+3cpq+lScq2+6bk2

+ 6¤5q+2b5k+apn2)+12p2n2

I — 35 — I~ 3- 3 9 9 Q I

+I3abn2+2OcdU2+hacq<2abkI

I äPW2)36ÜU2(?+2k)]/allI

I.5 Q [I 'Q n Q [I;(3°‘]_’)

LI, *" b&b}\.”X TI(·lOCL«^¤X’I‘2LI.b?a.1(.‘9I"l;,«-„>c»“»)(”I'S?-7'V] C1 '*

(3 ·«

(32l22)b * l2abL3X3U[Ök2X2(l5cd+2&b)*l2LX(l2Cq„

. , .3 « (1 7cp+bx>""95bk‘* &k‘*‘ab¤‘*)3‘*L»ß;<>#19“"‘l·8k@3‘2]/@1IE '-‘ 3

” „ ° „„~„(3‘(’3Öck‘~3aIb112~ccak)"I’(_>3«*¤X(äßcqr36<>q¤2‘*·3¤Jq+@¢p¤2+@~1¤‘¤*“ I

I(33. Izg Cab(“*2X2'VI($?*~·X’*‘l)/GfXI

r . .- t' _|_ ,= J-

I

(31 3)h Xesaba X [XX(4a·3bFLc«3¤)·l„w 3/G I, - ~ ~ I q 3(3l?22)h'*·l2abN5X5n[AX(5a*lOb+2ÖC-2a)*2gR2]/a I

I(313;* )}I 2lI,g_b?\.3 X? [$*62 xß (3bc;:-c1a)+?(.X( l+cI‘ I

I+„£éÖU2+2Cj3$&Ug+§K&+l2bW2)+T@k¤2%?ÖW2+j£]/@9 --

... ..

_ 3(32*)h = 18abk2X3¤[A2X2(a2+3ab+5bc*LOcd

- 5¤a)+AX(aJ+3ak+ban2+2nbn2

+ 6c3+12¤r+72cn2+18¤k)+36k¢ ef? 1 /,1 ~+ 232]/alo

Products _o_;_‘ Order Q

(36)}(3*12)}

= lO8a2b2k7X7/all

(3*22)}(321222)}= 12a2bmÖx6/eg(322*)}= 72a3bk7X7/all

(33l)}, (322)} , (3212)} • (33122)} , (33122)3 , (3313); , (3323)} ,(32132)} „ (32123)} , (312)} „ (31*2), (31322)} , (3l222)3 , (312*) „am (323)} = 0 ‘ 2

(;6)h = 3210 a2b2A6X6[A3X3(2Ocde+3dab)

+ 912x2n2(30cd+ab)+161x(3¤q2+bk2)+10p=n=J/136

1

.. ..

(3 52 )1* lOcd+a2 )+lJ\·X( 3cq'n

+ bk)+2kp(n+l)]/alö(6 *12)1*(6

*12)1*+l8k2X2(l3abn2+3Ocdn+6abk)

+ l8NX(3cq2+bk2)+bn2]/alß

(3 *2 2) 5x $[1 3x 3(20¤(1e+3¤1ab)

+ 912x2(30¤dn2+5abn2+12bcq)

+ 6LX(2lbk2+2Obpn2+9cq2+6jk)+6p2¤2]/alu

ll+ 2kX(ap+l8cq+6bk)+6kp]/a(32l22)h = 72abk2X2[3abkZX2(a+3c)

· + 3aAX(l3bU2+ÖbkU+2Cq+6k)+äPU(2&b+3PU)3/@12(32l2)h = 66ab1*x*[3ab12x2+161x(cq+6k)+10pnß]/alo

— 38

zoqabmhxhn/¤7zouabxdxhn/Q7

gscazbkßX5n(anx+8bmx+12¤nx+1on2)/dll

Products _9_;§_‘ _Q;·_dg_;_· 1

R ·· 39 ···(3Ö2)h = 58220a3b3kn9x9/@17

(3522)h Z h32O&3b2Ä8X8n(l+3bÄX)/@16

= 2Ü52 8,2b2Ä,7X71]/lalz

RBAZBRL = h32a3b2N8X8(l8k+j)/al5<3“122)h

= 2l6a2b2k7X7(akX+6k)/al3<3“122)h ¤ l728a3b2k8X8@/qlh(331“)h

= 108abpx5x5n/„l¤

Z Vßazbkéxönt6cnx+6b+„2)/G12

Bßoabmßxßn/G8R

(32lh”2)l" Z'-3Ö&b7\•5X5(2a7gX+3k)/Q_9(32l3l?„2)h

=-= lhhazbkéxén/alOR

RR

Ä Ä.. ..

0

Z 72ab7~.5X5·q/418Ä

Z (31323)h Z

zssazbxßxén/010

ixxßagsä Qi QEQQE

Ä

Z Bzhazbzksxs/G12

T

.. g)_]_ ..

A · P ··APPENDIX B(l)

Exgcted g·Products lgggggggg gl, gz, Q3 ggg gk Through Order Q

(Note: See Appendix A for notatiou)„

Products gi Order Q(h2)l

ZProductsgg Order Q

(n?)2 = 576abcunFx“/all(h?l)2 Z 96abcN&Xun/ag

<u22)2 = 96abcrnFx“/eg

<u22)2 = 2habckFXh/¤7(4l2)2 = 0

(hl3)2 = zqabcxäxh/Q7

(AZB)2 = lbhabckyXhn/ag

A

„ A3 -

(432)} = 2l6abcqkhX%hx9

Products gg Order 4

4 (42l2)2 = 24abc45x5/aß

(4222)2 = 48a2bcNÖXÖ/¤lO

(423)2, (433)2• (4122)2„ (4l23)2• (4l22)2„ <4223)2, (4l32)2„(4232)2; and (4l23)2 * O

(44)} = 576&bCÄhXh[2NhXh(35d6fg~G&C)

(4,31)} = ll52abc7~.4X4·n( 5dv7~.X+2cs7~.X+2t )/dl?

(4,33)} == 23O4„abc7~.4X4·n[?«.3X3(35def+abc)+37»2X2(lOdez

+ bcr)+34X(lOduv+3cqs)+6st]/dl4(42l2)3 =‘l92abcN4X4(dNX+2n2)/d9

(lvzßg)3

13

10ll

( 1+223

0

3 A)3 I

+ 31¤>q+l2qS]/¤·l2 II

III

IIIIII

IIII

Pmducts _g:§ _Q_r_(j_q_r; Q18

lßazgahbhchggxhwzglö

x2 ) ,/g hh

0h

(1+313)

10

....,

($3122)} 288a2bck6X6n/011

(12223)}2(1,2132)}2 lOO8a2b2c7u7X7n/al}(1,2232)}2($123)}

2 lle-’+<'ä2b2cÄ.7X7/[cx 12(111)} 2

0(u3”)}2 l29Öa2b2cqN7X7/alu(4122)} = 0(1.,123)} == 72abc7».2X2/018

(1123)} = 0

(1,233)} 2 801,a2b¤1((°>(Ö·q/011

(1,132)}(1.,233)} 2 2592a2b2cA.7X7n/0.13(1,122 2)} = 21,abc>(3)(3/03(1,12321} 2 21éabcq7x.5X2/010(1,2 23 2)} 12

I

— h? — I

(hl223)3 S h8a2bCÄÖXÖ/alg

0IÄSIÄ S 9ZEabchhXh {lOd@fgXhXh[lh7hRX(iRX+l2n2)

‘*‘ 579l’fIL"I 7O5h7\-XTI2‘*’5O1+O@?&»XTI2+*8L,OzI 2h7x.X

+ l)H2+Ö3d%X(eKX+4n2)]+LOd€fN3X3n2[882qRXIhRX

S lOm2)+3lO8nh+2lOz(3gAX

I+ Znh)+u8Ovyzn2+l0tz2+4b56cszAXn2

+ 3bcrzn2x2+g80vyzn2]+12¢xxn2[v2w+12uv2+ 582SuVKX+LbcrVK2X2+abcVÄ?X3]+3t[l2t2

+ l2b2c2r‘3Ä„L"XLI”I/ali}-l35l+8Ü&2b2CI3Ü7¤8X8/@18 I

(1.,.L”1)h 230L,ab¤;»„'+><’+n{5>».3><3[22z,d.e£g>xx+

1288defn2+28defNX-uabckX] I

I

I

I

II

- hg -

+ l2t[5dVkX+2t+2cskX]

+ l2cskX[deuKX+2sn2+3cqKX]

+ hbcrk2X2[5dAX+2r+3ckX]

+ abck?X3[5dhX+l6n2+hckX]}/alé

(Ah2)h Z ll52abckFXh{lLOd6fg[3kkX+8hn2+dkX]

+ 4Odefk?X3n2[39gkX+892n2+56r+2lcAX]

+ 2dezk2X2[225(fgk2X2+8fkXn2+2n2)+3Ot+ 2AOOyn2+hOrz+36ckXn2+l5bck2X2]

+ LÜÖVÄXQZ[9Ü€yNX+Ö5W+l2PV+l8U+l2t

' + Lrs+9cqkX+3bck2X2+©t]+bcrk2X2[l5deN2X2

+·l6OdNXn2+8r+l2t+72ckXn2

+ 6bck2X2]+8abcÄ?X3n2[5dkX+2r+3ckX]

+ Zurtg-llaabcrkhXh}/al?

+ 4n2)+3Oz(fgk2X2+8fkXn2+2nh)

+ 2LOVyn2+htz+2hcVkXn2+bcVh2X£

N- ÄQ -

+ 6cqzkX+7cfgN2X2+bcz]+lOdukX[A?X3(2lefg+abc)

+ l2kX(lOeyz+csu)+l2V(5w+t)]

+ 7Odefsk3X3(gkX+32n2)+8sK2X2(lOdez2

+ bcr2)+96skXn2(5dV2+cS2)+6cqkX[3OduVkX

+ 3st+9bcqskX+BbcrK2X2

+ gOdVkXn2+2rt]+2abck3X3[5dVkX

+ 2t+2cskX]+2LSt2—5aabckhXA}/alg

(h3l2)h * ll52abckhXh[k2X2(5dez+bcr)+NXU2(h5dVE

I(43l2)h = ll52abckhX“{khXb(35defg—aabc)

+ 3kX(lOduV+lOdV+3cqs+2cS2)+3St]

(4332)h = l728abcNhXh{2Odek2X2[lLfg(2hNX+l5n2)

+ 280fKXn2(3gKX+4n°)+3fkX(64n3+2lgNX

+ ll2qn2+lOgzkX+8Ozn2)+6OZ¤h

.. 5g ..

+ 24csXXn2··Y·bcr7¤2X2]·Y·6Od?x.Xn[l2Oeyz7»X

+ €>OVw+ l2tu·Y·l2 c su7».X·<·4b c7».2X2+abc?»3x3

Y ·Y· 24qV2”0]+3 qix.L*XL*(7Odefg—3uabc)

+ 24cqNX(l2s2n2+br2)+Spn2(3OduVkX

+ 3S’c+9cqS?\.X‘*3bcx·?¤.2X2*‘·abclY‘¤3X3)

+ l2b¤A.X[7~.2X2( 5dez+bc1~)+4Jx.X·q2( 1GdV

9cqs+4OdVn2+l2csn2+·2at)+3st ]/on lk

+ ÖOdukXn2(4u+3V)+ÖcshXn2(85+9q)[

Y 2x·t:+l2sX,?(n2+bcr?¤2X2)·Y··l2t2+l8sun2+l4abc?»3 x3 112 /ql5 k

Y(4323+

2lk+378n2)+l2ez7»X(2z·Y”3k·Y·lOy)Y

+ 4bcrk2X2+abck3X3]+36t(2t+ks)

— Sl — I

IL

LI

I

I

III

..52..

(#2123*h = 288abcÄhXhh[2abk2X2+5KX(br+4dV)

+ 14u+12kS+6¤n2]/alk

= l5dV+2an2)+l2s(j+6n2)+Lrn2]/alz

(h2223)h = 576abckhXhh[2Odek2X2(7fkX+8z)

+ 2OdVNX(eNX+37U2+2bNX)

+ LRX(at+27dt+ht+9cdskX+3asn2

+ 2brkX(l2ckX+AbNX+akX+8n2)

(h2l32)u L lAuabcAhXhn[LOdek2X2(lufkX+l5z)

+ l2OdVkX(2ekX+2ln2+3dhX)+l2t(27dkX

+ 26n2+2ckX)+72qs+kpn2+l2rhX(abRX

+ 8bcAX+3bn2)+uabN2X2(6ckX+9n2)]/alu

(42232)h L 288abcA#Xh{lAOdefN3X3(gAX+42n2)

+ l2OdVRXn2(lOe%X+l5u+l2q+6OAX+n2)

+ 12x2x2(5de¤+30¤dSn2+3absn2)

+ 36kX(2Odtn2+bkt+l2cqsn2+hbksn2)

..53...

+ l2brkX(3cqkX+2pn2+jk+RXn3)

+(42123)h

= 576abcx“X“{m2X2(abr+5abn2+1Odez)

+ kX(llOdVn2*£at+9dt+6asn3

+ &8csn2+3br+Abrn2)+3Otn2+18snh}/al}

> h8abcN“Xh[k2X2(l5de+6bc-Zaa)

<n3“)n “#32abcRhXh{lO§d@fN3X3(d%X+2qNX+27U2)

+ 6OdukX[5ekX(fkX+4n2)+5Ln2(2eLX+n2) [+ 6qu+3kpn2]+72s¤2[k2X2(ab+5Od6)

[+ ékX(bk+l5dn2)+3q(Ls+p)]

Bqflocdmzxz(26mX+27n2)+9AX(abxXn+

lOcd)+LAX(3cq+bk)+2kp] Ä

+ éabA2X2(2cqxX+pn2+bkAx)}/al5

zléabcxhxhn/agI

T

— 5h “

(hl33)h = 2habc%hXh[2kX+9qn+l9n2]/ag

(Al23}h = 9ÖabcNhXhn[kX(lO2a+lOl)+A6n2]/alo

(h233)h = 96abcRhXhq[2k2X2(L5d6+9bc+a2+hab—3¤a)

+ NX(l80du+2LbS+L8ds+9aq+36bq

+ lO8cq+72bn2+l2an2+2an2)+228sn2

T+l8cNXn2+l29nh+6qu+bkKX)

+ ZABXB(lOcde-aab)+9k2X2n2(3Ocd+ab)

(4233)h “h32abckhXhh{l5dek2X2[kX(35f+l2d+Lc)

+ 6NX(l5dn2+bk)+3q(bs+p)

+ ZE?k2X2n2(ab+3Ocd)+l8kX(3cq2+bk£)

+ ukX(3cq+bk)+2kp]}/ulh

(Al222)h =

V...55...

lhhabchhXhU[75deN2X2+2OdkX(2eNX

(

- 56 -

APPENDIX B(2)

* Expected Q—Products of Order 6, Z and 6

(Note, see Appendix C for the evaluation of necessary expected

q-products- All expected Q-products not shown are O).

Products Qi Order Q

e (ré)3 * l5[r2]3, (rhs2)3 = 3[r2]2[s2], (rzsztz)3 * [r2][s2][t2]

(r6)4 Z lO[r3]2+l5[r2][rh]—h5[r2]3

(r5s)h ~ lO[r3][r2s]+l0[r3s][r2]

( (rhs2)u = 6[r2s]2+h[r3][rs2]+[rA][s2]+6[r2][r2s2]—9[r2] [sz]

(r3stu)h = 3[rtu][r2s]+3[rsu][r2t]+3[rst][r2u]+[r3][stu]

+ 3[r2][rstu]

(rgsäh(r2e2t2)h — tlreu]2+2[r2s][su2]+2[ru2][re2]+2[r2t][e2u]

+ [r2t2][s2]+[r2s2][t2]+[s?t2][r2]8[r2][s2][t2]

* This Appendix is taken from Shenton and Myers [ll]

}

(r f s f t)•

...57..

(r2s2tu)h L ulrsu][rSu]+2[r2sJ[6cu]+2[rS2][ruu]+ [r2u][s2t]+[r2t][s2u]+[r2tu][s2]+[r2][s2tu]

·(r3s2t)h = 3[rsä[r2c]+6[r26][rSz]+[r3]{s2u]%k2J[r3u]+ 3[r2][rS2c]

Pp¤duccs gi Qgggg Z(r7)h L 1¤6LLL12LL31

(rés)h L usurguzcrzsn_ (r5st)b L

L<r“SLu>„L 3[r2]2[stu]<r“S=¤>b L 6£rLJ£S=JLrL¤J+3£rLJ2ESL¤J

L(r3s2tu)hL3[r2][s2][rtu](r3s3t)h= 9[r2][S2][rSu]

>

(r2s2t2u)h L [r2u]E62][u2]+[c2u][r*][s2]+[s2u][u2J[r2]1

<- 58 — 3Products 3;;) Q_;_·_c_1§_; 3

<1~2S·%*=u2>,+ = £1~2J[S=JEv’=J£u2J

..

APPENDIX 0

Supplement to Appendix Bggl

(Note: See Appendix A for notation)

[12] 2 AX/Q[22][32]

= éabnßxg/a5

=·[33]Z 576abctkhXh/all

[122}[123]==0==0 [1a2]· 96abckhX2n/ag

[223] Z 2aabh2X3n/06 [232]~ 36abkA3X3/07 ][22b,]

2l6abcqkhXh/ag 576abcsk2Xhn/alo

0

[134]

1

... ÖO ..

[132] ¤ éanzxßn/ab ‘

3[133][131] = 0[231][233][23uJ L[331] 1[332][331]

3 132abckhXhn[k2X2(ab+2Ode)+6LX(5du+bk)+ 3q(p+hS)]/alz

[131][132]

[... 61 ..

ggägj

lOdez)+37»X(lOduV+3cqs)+6st]/alu

[1222]2[1232]

2 18ab43x3(2¤4x+3n2)/a7[1242] 2 24abckhXh(akX+8q)/ag[2232] 2 12abk3X3(aaA2X2+l8cqkX+6akXn2+l2pn2+6jk)/ag

[2242] 2

2[1223]23Oabh3X3n/a6[1224]2 24ab¤42x#/47

[1234] 2 l68abckbXhn/ag[2213]2[2214]

2 l92abckÜXhn/agl[2234] 296abck#Xhn[kX(a+6b+27d)+27¤2]/alo[3212][3214]

2[3224]2l44abckÜXh(abN2X2+lOdukX+l2bkXn2[4212]

2 l92abckhXhn(akX+lldkX+6n2)/alü [

[[[

..62+

ang]/all[1;,223] l92abc?x.l*Xl‘Tl(3ab7«.2X2+3OdV7&X+6br7~.X

+ l8t+l8sk)/alg[1;,123] =·· 21;abc7«.,"'X,+(2a7»X·+·9d7»X+3O1‘;2)/019 1

ß1 1

l

AQQNOWLEDGEMENTS P4

, gg -

REFERENCES

[1] Anscombe, F. J. (1950) Sampling theory of the negativebinomial and logarithmic series distribution.Biometrika, jl, 358 — 362.

[2] Bartko, T. J. (1961) The negative binomial distribution.A review of properties and applications. EggVirginia Journal Q; Science, l2(1), 18 — 37.

[3] Bowman, K. (1963) Higher moments of maximum likelihoodestimators. Thesis for Ph.D. degree, VirginiaPolytechnic Institute.

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Y _ _

1

ABSTRACT

This thesis is an extension of the development oforthcgonal statistics which can be used to investigatesampling properties of moment estimators. This work is

particularized for estimators of parameters of the negative

binomial distribution.If we call qr(x) (r=O, l, 2, ...) the set of orthogonal

· polynomials associated with the negative binomial distri-bution, then the rth orthogonal statistics is defined as

__ *2*

The thesis contains tables of the expansions of expected

powers and products of the Q*s in terms of the populationparameters for the parametric form due to Anscombe [1 J,

1through terms in n”h. These products and powers involve 1

Ql, Q2, Q3 and Qh. Since the Q's are expressed as linearfunctions of the sample moments, one can expand a momentestimator t (or power of t) as a linear function of the Q*s,and then through the use of the table of expected Q—products,evaluate such properties as bias, variance, and higher order

moments of t, as functions of the population parameters. Abrief example of the use of these tables is given.