This article was downloaded by: [University of Chicago Library]On: 30 November 2014, At: 02:45Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Communications in Statistics - Theory and MethodsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lsta20

Asymptotic relative efficiency of some joint-testsfor location and scale parameters based on selectedorder statisticsA.K. Md. Ehsanesh Saleh a & Pranab Kumar Sen ba Department of Mathematics and Statistics , Carleton university , Ottawa, CANADAb Department of Biostatistics , University of North Carolina , Chapel Hill, H.C., U.S.A.Published online: 27 Jun 2007.

To cite this article: A.K. Md. Ehsanesh Saleh & Pranab Kumar Sen (1985) Asymptotic relative efficiency of some joint-testsfor location and scale parameters based on selected order statistics, Communications in Statistics - Theory and Methods,14:3, 621-633, DOI: 10.1080/03610928508828938

To link to this article: http://dx.doi.org/10.1080/03610928508828938

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shall not beliable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilitieswhatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out ofthe use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Deparrmen-c uf k l a t h ~ m a t i c s and S t a t i s t i c s C a r l e t o n Univers i ty Ottawa, CANADA

Key NO&& t; P b e n : ABLUE; anymp&Ac o p t i m a L ~ y ; a,lympto-tic x e L a t i v e ed&cierzcy; BLUE; g e m a L i z e d L e n n t n q m e ~ ; _ae m d i z e d v a n i a YEP_; RocaRio c d e wu d d ; qua nLLte ~ u m t i a n; b p a c i q n ; nymmdhic d i h t h i b d i u n.

ABSTRACT

For a l o c a t i o n - s c a l e d i s t r i b u t i o n , based on few s e l e c t e d o r d e r s t ~ t i s t i c s ~ . - i o i ~ l t - t e s t s f o r t h e I n r a t i o n and scale parameters a r e cons idered . I n t h e l i g h t o f t h e asympto t ic r e l a t i v e e f f i c i e n c y r e s u l t s , op t imal spac ings o f t h e o r d e r s t a t i s t i c s a r e s t u d i e d (and t h e i r d u a l i t y t o t h e e s t i m a t i o n problem i s d i s c u s s e d ) .

1. INTRODUCTION

L e t X n , l < ..- < X be t h e o r d e r s t a t i s t i c s of a sample n:n

of s i z e n randomly drawn from a l o c a t i o n - s c a l e d i s t r i b u t i o n F ,

where

F i s a s p e c i f i e d d i s t r i b u t i o n f u n c t i o n ( d - f . ) w i t h a cont inuous 0

p r o b a b i l i t y d e n s i t y f u n c t i o n (p.d.f . ) fo, and 0 and y (> 0) a r e

unknown l o c a t i o n and s c a l e parameters . Le t Q0 = CQ ( t ) , 0 < t < l ) , 0

t h e q u a n t i l e f u n c t i o n , d e f i n e d by

Copyright O 1985 by Marcel Dekker, Inc.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

~ e t z = X . , X ) ' he a v e c t o r of k ( 2 1) s e l e c t e d -n n : r

I k s t a t i s t i c s w i t h t h e ranks r

11---' rk, determined by t h e spacing-

v e c t o r X = (X . = = = ; X ! ' bv l e t t i n g - 1' k -

where [s] i s t h e l a r g e s t i n t e g e r con ta ined i n s . We l e t t h e n

= 1 , . , , 1 * and 5 = 81 + ya. - - (1.6) - Then [viz . , M o s t e l l e r (1946) 1, whenever f (a,) > 0 , 1 j 2 k , a s

0 I n i n c r e a s e s

Note t h a t a and a r e known, b u t 5 and y a r e unknown, Ogawa - (1951) [see Sarhan and Greenberg (1962)] i n c o r p o r a t e d (1.4)-(1.7)

i n -&e foL?ulation of a s y m p t o t i c a l l y b e s t l i n e a r unbiased

e s t i m a t o r s (ABLUE) o f ( 8 , y ) , based on Zn, which a r e g iven by

where

- 1 kl j l l Y 1, k12=k21=a'~-11, k 2 2 = ~ 1 ~ - 1 a ( n o n s t o c h a s t i c and known) - - - - - - - - -

c1.9)

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

ASXQTOTIC RELATIVE EFFICIEXCY 623

IY = Z and M = a ' ~ / ~ z (stochastic) (1.10) I n - -. -n 2n - .. -n

and hence, by (1.71 and (1 -11) we have

r , e t 2 = [ [ I . . ) ) be t h e Fisher ( loca t ion-sca le) information matrix - - - - 1 3

of F (i=e= of F when 0 = 0 , y = 1). We assume t h a t 1 i s a 0

f i n i t e , p o s i t i v e d e f i n i t e (p.d.j matrix and denote i t s detaLqinant

by / i ! . Tben, t he ABLUE of (0 ,y) ' based on a l l t he n order - -

s t a t i s t i c s [viz. Ang (1955)1, denoted by (8' y j ' , has the n n

following property [viz., Chernoff, Gastwirth and Johns (1967) 1:

d i s t r ibu t io r

Note t h a t 5

To study the asymptotic r e l a t i v e e f f i c i ency (A.RE.) of A A

(en, Y ) with r e spec t t o (8,, y n ) , we make use of (1.12) and (1.131 n

and employ the generalized var iance of t he two asymptotic

IS, which leads t o the A.RE.

depends on @ and (and hence, on and f 0 ( ' 1 ) . - Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

,o, For a given p.d. f. f and k (2 l) , l e t us d e f i n e f = i ( 9 , . . . ,*,I

0 - h. r

responding t o a given A. This cho ice ( A " ) of t h e spac ings of t h e - o r d e r s t a t i s t i c s l e a d t o a maximum of t h e A.X.E. i n (1.14) . For

77-w. ,,,;,,, ,..>" F and s p e c i f i c k (2 I ) , t h e optimum choice o f h0 has been 0 -

s t u d i e d (numer ica l ly ) by v a r i o u s workers [ v i z . , Sarhan and Greenberg

(1962) and Kul ldor f f (1977) 1.

We a r e p r i m a r i l y i n t e r e s t e d h e r e i n t h e ( d u a l problem) j o i n t -

t e s t of

based on Z , where 8 y a r e s p e c i f i e d . i n t h i s c o n t e x t too , o u r - n 0' 0 main emphasis is on t h e A.R.E. r e s u l t s and on a s y m p t o t i c a l l y o p t i -

mal s p a c i n g s a s wel l . The proposed t e s t s and t h e a l l i e d

,: -L. -21 . . .L: _ _ L L _ _ _ _ _ _ _ --.-, ,-,7T+?:--J : - C ~ C + ~ , - , rnh- ,,ir: re=?:!+.; i i i a i i i i i u i i i i i i ;;u;';i; -,, ,,,,;;;c; A;; -,,,l-., -. -..- ------- on A.R.E. and op t imal spac ings a r e p r e s e n t e d i n S e c t i o n 3. Some

g e n e r a l remarks a r e appended i n t h e concluding s e c t i o n .

2. THE PROPOSED JOINT-TESTS

Cons ider f i r s t t h e q u a d r a t i c form

By (1.12) and t h e Cochran theorem, we conclude t h a t under H 0

o r Ln h a s a s y m p t o t i c a l l y t h e c e n t r a l c h i s q u a r e d i s t r i b u t i o n w i t h 2

degrees o f freedom. When H does n o t h o l d , b u t H does, 0 A

-1 0 - 2 2 n Ln converges i n p r o b a b i l i t y t o yo {k (8- 8 ) +

11 0 2

2k12(0-00) (y- yo) + k (y- yo) I = A > 0 , s o t h a t iO tends 22 . n

t o m ( i n p r o b a b i l i t y ) , a s n + T h i s s u g g e s t s t h e use of

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

rhe right hand side tail of the discxibution of LO as t h e 0,

appropriark critical re:i=~; tke critical value (of 1 , , I1 , E li

~orres~urldinj r o G s;czified o i g n i ficance level E (iic E < 1) , - - - 7 ' " ' n P C LVfi , --2-- x ~ ( E ) , the cpper it3024 p o i n t sf t.+ie ccntral

chisq~are d.f. with 2 DF.

Along with this asymptoric test, we like to c ~ n s i d e r ,

side by side, another verslon, wnere insiead of die asynptotic

momenis of Z , thc exact ones are used in the construction of the -n

test statistic. For this purpose, .VIE define for every n (2 .- k i ,

For various F and specific (n,k), '3 and V have been 0 - n -n

comp~dted and tabulated [viz. Sarhan and Greenberg (19623 1. Then,

by the generalized least squares (GLS) principle, the BLUE of

( 8,y) based on Z is given by -, n

where

Parallel to (2.11, we consider then tne test statistic

0 * (2.6) To study the affinity of L and L we denote by

n n

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

Then, we have by 12.3)-(2.5) and (2.7);

so that by (1.11) and (2.8),

At this stage, we make use of the moment-convergence results of

sample yuaniiies [see, for example, Sen (1959); and conclude that

under tne assumed regularity conditions,

so that by (2.9)-(2.12), A

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

Havinq obtained t h l s asymtotic equivalence, we conclude that &

L i has asymptotically the same distribution (under H ) as 0

LO .

To find the asymptotic distribution theory under HA in (1.16), n

we coilsider a seyueilce of alternatives {A f , where under A , n n

where 6 = (6 6 ) ' ( f 0 ) is some fixed vector. Using (1.12) , 1' 2 -

!2.15), (2.11 and the Cochran theorem, one aqain claims that

under {A j , LO (or L* ) has asymptotically non-central n

chi square distribution 2 DF and non-centrality parameter

- - Side by side, if we would have used the ABLUE (8 ,

n Yn) in (1.13) and a quadratic form in this vector (as in (2.6)),

- then we have a test statistic L- (based on the entire set of

1,

order statistics) which, under (A_} , will have asymptotically I I

non-central chisquare distribution with 2 DF and non-centrality

parameter

where ? is defined after (2.12) .

3. A.R.E. RESULTS

0 TO compare L (or L* and , we note that the classical

n n n Pitman-A.R.E. result is applicable here (both having the same

asymptotic size and similar non-central distributions), so that,

we have by (2.16) and (2.17),

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

Vnl ike t h e c a s e of unigarameter t e s t s : h e r e (3 .1 ) depends: I n

g e n e r a l , on t h e d i r e c t i o n - c o s i n e s of 6 , s a t h a t ( 3 . 1 ) i s a - f u n c t i o n of K I and v i where 6 = 6 cos 1) and 8 = 6 s i n v - - 1 2 ( 0 2 V 2 2 n ) . One may t h e r e f o r e be i n t e r e s t e d i n s tudying t h e

range of v a r i a t i o n of (3.1) over t h e range of v a r i a t i o n of V.

Note t h a t by t h e Courant theorem on t h e extrema of t h e r a t i o

of two q u a d r a t i c f o r m [ c ; f = R.ao (1973, p ; 5 (? ) ] ; we have

0 - -1 i n £ A.R .E . (1 , i j = e i K i I ) = ch ( K 1 ) = e say ( 3 , 2 ) V 0 - - 2 -- 2

0 -1 SUP R.R .E . i ~ O , i ) = e (K,I) = ch ( K T ) = e , say - - 1 -- 1

( 3 . 3 ) v

wnere cn ( a ) and ch ( ' 1 s t a n d f o r t h e l a r g e s t and s m a l l e s t 1 2

c h a r a c t e r i s t i c r o o t s . Thus.

and % = K T - ' / ' - - , ( 3 . 5 )

With t h e s e r e s u l t s i n hand, we proceed now t o c o n s i d e r t h e

a s y m p t o t i c a l l y op t lmal cho lce of: t h e spac lng A, f o r a given - k(, 2) and F . - 0

(i) Max-min o p t i m a i i t y : For a g iven k

A , i n such a way t h a t f o r t h e corresponding - and F we choose

0

K ( = K ( A ) ) ,

Thus, t h i s op t imal cho ice of r e l a t e s t o t h e maximization of t h e

minimum A.R.E. (over V €[0,2n]) i n ( 3 . 2 ) .

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

( i i j fi-oUtiTi-?:L... I a i r i v . Note t h a t r h e Gaussian c u r v a t u r e of

t h e asympto t ic power f u n c t i o n of 1': under ] i n ( 2 . 1 5 ) , n

a t t h e null p i n k ,3 = 3 , is maximize2 W ~ E E i s a maximum,

4 . e . , 1 KT-' i s a rna:<i~:Grn ( a s 1 does n o t &pen6 on A , b u t -- - - K does s o ) . By (3 .5 ) , t h i s amounts t o maximizing - 7 4 - 1,

Je e = - ( j j, / I , j '? by ttle d e s i r e d ~ h c . i , c ~ _ of ). . For 1 2 -- - - -

t h e D - o p t i m a l i t y , we may r e f e r t o I saacson (1951) , f o r a very

e l a b o r a t e discussion. i n t h i s s e n s e , ~ i i e opilitiai cho ice of

X a g r e e s w i t h t h a t i n (1.14) -( 1 . l 5 ) , and hence, t h e d i s c u s s i o n - fo l lowing (1.15) a l s o p e r t a i n s t o h e r e .

(iii) Trace-op t imal i ty : We conceive of a p a r t i t i o n of t h e

parameter space of (under ( 2 . 1 5 ) ) by means of t h e e l l i p t i c a l

con tours :

where H i s d e f i n e d by (2.17) . Note t h a t i = c <=> 6 ' I6 = c y i = c* * - --

( O - < c < m! On t h e s u r f a c e of t h i s e l l i p s e we a s s i g n a uniform - , - - p r i o r d i s t r i b u t i o n of 6 . Note t h a t on a -contours , t h e a s y n p t o t i c

G * - power of L ( o r 1 ) may depend on V . I n t e g r a t i n g o u t (3 .1 )

7 'l

w i t h r e s p e c t t o t h e uniform p r i o r d i s t r i b u t i o n of 6 on a - A-contour, we o b t a i n an average (over L') A.P.E., and t h i s reduces

t o t h e t r a c e c r i t e r i o n :

Thus, one may choose t h e spac ings 1 i n such a way t h a t (3 .8 ) i s - a maximum i . e . , we have an a s y m p t o t i c a l l y op t imal average A.R.E.

Note t h a t does n o t depend on t h e cho ice of 5 ( b u t , K does s o ) , - and hence, we need t o choose 1 , i n such a way t h a t -

1 , , d l 1 22 -[i- in, r i 2k (A) 112 + 011 1 = maximum 2 11 - 12 - D

ownl

oade

d by

[U

nive

rsity

of

Chi

cago

Lib

rary

] at

02:

45 3

0 N

ovem

ber

2014

SALEH AND SEN

I n g e n e r a i , t h e s o l u r i o n s i n (i) , ( i i j and (iii! a r e nor

rhe s a x e . Ir, a rna3orir.i of tne c a s e s , r i s a s y i m e t r i c 6. f . , 0

~ . e . ,

s o t h a t

(3.12)

Moreover; by irn.posing i n v a r i a n c e and symmetrv on t h e BLUE; T,qe

may r e s t r i c t o u r s e l v e s t o t h e c l a s s of A f o r which A + A ,=l, -" i k + l - I

f o r every j = 1 , 2 , , . . , [ (k+1) /2], and t h i s would imply , -by v i r t u e

of ( 1 . 4 ) , (1 .51 , (1.9) and (3.10) t h a t k = k = 0. So t h a t 12 2 1

(3.121 reduces t o

AS such (i) Maximum o p t i m a l i t y i s ob ta ined by maximizing -1

k22722 over X .

(ii) D-optimali ty i s o b t a i n e d by maximizing / k l - l l over A, - - and

( i i i ) Trace-op t imal i ty i s ob ta ined by maximizing k 1-1 + 11 11

k I-' over . 2 2 22 D

ownl

oade

d by

[U

nive

rsity

of

Chi

cago

Lib

rary

] at

02:

45 3

0 N

ovem

ber

2014

rhem wlrh r h r e e examples.

(a! Normal case; Here i = ii i = 2, 1 = 1 = 0. 11 . . 2 2 21 12 -. 1 7. ,Inen, for maxi-mi? 3 p t i i ~ , a l i t y we maximize - 5 Pis r e s u l t s

2 22 a r e a v a i l a b l e i n Oqawa (1951) [ v i z . s e e Sarhan and Greemberg

1962 T a X e 10 .I2 - 4 1 . The r?a:*_,rr.&~ efficiencies f o r k = 2 (2) 6 a r e

f o r t r a c e op t imal l t i ; we maximize k + L k . 'Thc op t imal 11 2 22

spac ings d i f f e r from t h a t of D-optimali ty r e s u l t s and t h e

e f f i c i e n c i e s f o r k = 2 ( 2 ) 1 0 are ,6407, .8286, .9009, .9359 and

.9552. A l l t h e t h r e e o p t i m a l i t y r e s u l t s a r e d i f f e r e n t i n t h i s

c a s e

(b j c a u c h y ~ : Here 1 = i = ' i = i = O . Then, 0 11 22 2' 12 21

f o r maxi-min o p t i m a l i t y we maximize 2k9?. This has been done L. L

by Chan (1970) . For k = 2 ( 2 ) 10 , t h e e f f i c i e n c i e s a r e .8106,

4kllkl2. For k = 2 ( 2 ) 1 0 , t h e e f f i c i e n c i e s a r e -4677, .7659,

.8735, .931? and ,9469, and f i n a l l y , f o r t r a c e o p t i m a l i r y w e

maximize k + k which maximizes k k ( s e e Cane, 1 9 7 4 ) . li 12 11 22 - - ~. r u r k = 2 ( 2 ) 1 0 t h e e f f i c i e n c i e s a r e .6840, ,8752, .9346, .9600 and

.9732. The D-optimali ty and Trace-op t imal i ty g i v e t h e same spac ing

and t h e A m i s unique.

( c ) Double-exponential F : Here = I =1, 7 = I =o, 0 I11 22 12 21

Then f o r maxi-min o p t i m a l i t y , we maximize k This h a s been done 22'

by Cheng (1978) . For k = 2 ( 2 ) 10 t h e e f f i c i e n c i e s a r e .6476,

.8202, .8910, 9269 and -9476. For D-optimali ty , we maximize

k k The r e s u l t s may be o b t a i n e d from Cheng (1978) . 11 12 '

For k = 3 ( 2 ) 1 1 , t h e e f f i c i e n c i e s a r e -6476, -8202, .8910, .9269

and .9476 and f i n a l l y f o r t r a c e - o p t i m a l i t y we maximize k + k 11 1 2 '

The average e f f i c i e n c i e s a r e then .8238, .9101, .9455, .9639,

.9738. The optimum s p a c i c g s may be ob ta ined by us ing t h e r e l a t i o n Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

between optimum spac ing of doulole-exponential d i s ~ r i b u c i o n and

t h e e x p o n e n t i a l d i s t r i b u t i o n given by Cheng (i978j.

BIBLIOGRAPHY

Cane, G . J . (1974) . L inear e s t i m a t i o n of parameters of t h e Ca7dchy d i s t r i b u t i o n based on sample q a n t i l e . J&S-&, 69, pp. 243-245.

Chan, L . K . ( 1 9 7 0 ) . L inear e s t i m a t i o n of l o c a t i o n and s c a l e paramctcrc, of t h c Cauchy d i s t r i b u t i o n based on s&v,ple q u a n t i l e s . JASA, 65 , pp. 851-859.

Cheng, S . W . (1978) . L i n e a r q u a n t i l e e s t i m a t i o n of parameters of d m b l c - e x p o n e n t i a l d i s t r i b u t i o n . Soochow Zour. Math., 4 , pp. 39-49.

Chernof f , H., G a s t w i r t h , J . L . , and Johns , M.V. (1967). Asymptotic d i s t r i b u t i o n of l i n e a r combinat ions of f u n c t i o n s of o r d e r s t a t i s t i c s w i t h a p p l i c a t i o n s t o e s t i m a t i o n . Ann. Math. S t a t l s t . 38, pp. 52-72.

I s a a c s o n , S.L. ( 1 9 5 1 ) . On t h e theory of unbiased t e s t s of s imple s t a t i s t i c a l hypotheses s p e c i f y i n g t h e v a l u e s of two o r more parameters . Ann. Math. S t a t i s t . 22 pp. 217-234.

Jung, J a n (1'355). On l i n e a r e s t i m a t e s d e f i n e d by a cont inuous weight f u n c t i o n . Arkiv f u r Mathematik ad . 3 , n r . 1 5 , pp. 199-209.

K u l l d o r f f , G . (1977) . Bib l iography on o r d e r s t a t i s t i c s . I n s t i t u t e of Mathematics and S t a t i s t i c s . U n i v e r s i t y of Umea.

Rao, C.R. ( 1 9 6 5 ) . L inear S t a t i s t i c a l I n f e r e n c e and Its A p p l i c a t i o n . John Wiley & Sons I n c . , N . Y .

Sarhan, A .E . and Greenberg, B.G. ( 1 9 6 2 ) . C o n t r i b u t i o n s t o Order S t a t i s t i c s . John Wiley & Sons I n c . , N . Y .

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14

ASYMPTOTIC RELATIVE EFFICIEKCT 633

Sen , P.K. (1559). On moments of s m p i e q u a n t i l e s . C a l c u t c a S t a t i s t i c a l A s s t n . B u l l e ~ i n 9, pp. 1-9.

Dow

nloa

ded

by [

Uni

vers

ity o

f C

hica

go L

ibra

ry]

at 0

2:45

30

Nov

embe

r 20

14