kAD-R1?S 454 ONi THE LEAST FRVORABLE CONF IGURAT ION OF A SELECTION 1/1PROCEDURE BASED ON RAIICS(U) PURDUE UNIY LAFAYETTE INDEPT OF STATISTICS S S GUPTA ET AL. JAN 97 TR-S?-2

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u J. A" FAVOKABLE! UUNF UK'rJIU UF-A

SELECTION PROCEDURE BASED ON RANKS

by

Shanti S. Gupta*Purdue University

Takashi MatsuiI~Jm Purdue University and Dokkyo University

L Technical Report#87-2

00

PURDUE UNIVERSITY

~AWNWSELECTE,OZ MAR 3 1987I

DEPARTMENT OF STATISTICS

rris document hat be approved

fox luUc z*lele and sale; itsdistii~bu, ~is unlimited_

30 039

ON THE LEAST FAVORABLE CONFIGURATION OF ASELECTION PROCEDURE BASED ON RANKS

by

Shanti S. Gupta*Purdue University

Takashi MatsuiPurdue University and Dokkyo University

Technical Report#87-2

Department of Statistics ,. Purdue University .

January 1987

m..

'."

This document has been approvedfor public release and sale; itsdistribufon is unlimited.

, The research of this author was supported in part by the Office of Naval ResearchContract N00014-84-C-0167 at Purdue University. Reproduction in whole or in part ispermitted for any purpose of the United States Government.

Fr'

ON THE ASYMPTOTIC LEAST FAVORABLE CONFIGURATION OF

A SELECTION PROCEDURE BASED ON RANKS .

Shanti S. Gupta* Takashi Mataui

ABSTRACT

\L

The problem of selection of the population with the largest parameter is

considered using the subset selection as veil as the indifference zone approach eopyINS PEc 1

6for distributions that belong to a location or a scale parameter family. The

procedures are based on the sums of combined (Vilcoxon type) ranks and vector

(Friedman type) ranks. The least favorable configurations are obtained in an

asymptotic framework under certain order relations between the gaps of

parameters. The asymptotic theory is based on exact moments of the rank sum

statistics. A. ) " /

1. INTRODUCTION ILet i. 172 , .... I1k be k independent populations, where 111 has the associ-

ated cumulative distribution function (c.d.f) G (y), i = 1,2,...,k. It Is

assumed that (G ) Is a location or a scale parameter family, i.e. G0 (y) =

G(y -9), -oo < 9 < oo or GO (y) = G(y/), 0 >0. Let the orderedigi be

denoted by 0el ] & 0 [2 k . The population associated with O[k] Is

defined to be the best. Our procedures for selecting the best population are

Depertment of Statistics. Purdue University, Vest Lafyette. Indiana 47907.U.S.A. The research of this author was supported In part by the ONR ContractN00014-84-C-0167 at Purdue University.

5 Faculty of Economics, Dokkyo Unoversity. 800 Sakae-cho. Soka-shi, Saltama 340.I1.

based on ranks of observatioms from these populations in the location parameter

mae, ad am ranks of the absolute values of the observation in the scale Pa-

ranter ease. For my obeervatiom I from 6@ (P) In the sale rameter cane. the

c.d.f. of X a IYI Is Fe ) G(x/O) - G(-/9). For convenience of presents-

tion, we use FW() to denote the e.d.f. of IYI in the scale parameter case as

vell as that of Y Itself In the location parameter cane [i.e. F (x) G(x-G)J.

The we have

F66 (x) A F@6 (z) ' ... a F66 (x) (1.1)

for all x. Ve assum vithout loss of generality, that 1Tk is the best population

(I.e. Ok a 01. I a 1,29...,k-1).

Let YII' Yi2' .... Yin be n independent observations from I a 1,2.....

k. Let XI* a YI in the location parameter case. and Xiu = 1YIjI in the scale

paranter case. Ve consider rank sum statistics based on combined (Vilcoxon type)

ranks as well as vector (Friedman type) ranks. Let I denote the rank of X

(2)anong all kn observations In the cobined sample and kii denote the rank of Xii

geom X1j. %. .... Xkj. In ranking observations In a goup. the smallest Is

gIven rank 1. For a = 1,2. define

(a C n (a).

and

(a ) (a) (a) .... (a) 1.3)

bAere ca, a /n and cra 1. 1.

For selecting the best Population. we consider both the subset selection

approach of Gupta (see Gupta and Pauchaukesan (5]) and the Indifference zone

2I'.

2 .

'.

approach of Bechhofer (3]. For selecting a subset containing the best. we define

the following procedures:

(a) (a)R(a0 .1) : Select 11, If and only ifi (ia A mx - d

i = 1,2. .... k; do - 0; a,3 = 1,2. (1.4)

Here P = 1 and 2 correspond to the location and scale cases, respectively.

The use of these rule is Justified by Theorem 4.2. A correct selection (CS) Is

said to occur if and only If the best population (in our case f~k) is included in

the selected subset. Our aim Is to select a subset satisfying

inf P(CSIR(a.P.1)) a P* (1.5)

where a.P = 1.2; 1/k < ( 1; 1; = loll 02- 10. 'Pie E i = 1-

2,....k. 0 is a real line. The constant do Is the smallest nonnegative number

satisfying (1.5). the so-called P*-condltion.

Using the indifference zone approach, we define the procedures:(a)

R(a, P .2) : Select the population associate with H[k ] as the best.(1.6)

In this case, the rule R(a. P .2). . P = 1.2 are required to satisfy the

following probability condition:

P(CSIR(a.P .2)) a P* whenever -P. ( k. 0 i ) A cP 4 8 (1.7)

where a .= 1,2. l/k < P* < I. sp* > 0 is a given constant.

0 i -9J when P = 1

S i/Oj when P = 2 (.

and

0 when A -1cO o (1.9)

I when A a2.

3

Selection procedures (using both subset selection and indifference zone

approcbes) baed on this statistic H(1) have been studied by many authors

Including Lehmsa [9], Bartlett and GovindaraJulu (2], Gupta and McDonald [4],

Purl and Purl [15.16]. and Alan and Thompeon [1]. Also procedures based on H(2)

have been studied by McDonald [12.13]. MItsui [10] and Lee and Dudvicz [8].

A review of procedures based on ranks is given by Gupta and McDonald (6], and

Gupta and Panchapakesan [7].

A parametric configuration which gives the influ of PCS. the probability

of a correct selection, is called the least favorable configulation (LFC). It is

fairly difficult to establish the LFC for both rules R(a. P .1) and R(a. A .2)

using statistics H(l). H(2) and is still an open question in general (a A = 1.

2). The counter examples of Rizvi and Voodvorth [17] and Lee and Dudevicz (8]

show that the configuration 81 a 02 *** . k In the case of subset selection$

rules, and the configuration 61 a 02 . k-1; P ( *Ok. ek-1) Z cA + S

in the case of indifference zone procedures, do not yield, in general the Inflau

of the PCS. A discussion on the LFC can be found in Gupta and McDonald [6].

Our purpose in this paper is to discuss the LFC in an asymptotic framework.

An order relation Is assumed to hold between the "gaps" of parameters. This

assumption is similar to those considered by Puri and Purl [15.16). and Alas and

hoepson (1]. The LFC's of the procedure are studied by using the exact moments

of the combined and the vector rank statistics 1 (a). a 1.2. for location and

scale parameter cases (A 1.2) for both subset selection and Indifference zone

approaches.

4

8 mmelI

9 .~b ~ w j % , ' * 6 *

In Section 2. asymptotic distribution of U (a), a 1,2. are considered

under the assumtion of order relation between gaps of parameters. The PCS and

LC are investigated in Section 3. Moments results are given in Section 4 as an

appendix.

2. ASYMPTOTIC PROPERTY

2.1 Moments of Ranks

Let us define the mean vector and variance-covariance matrix of H(1) by

(1) (1)Pff# and A according as ye are dealing with location (.8= 1) or scale

(P = 2) parameters. Under the population model we considered in Section 1. the

elements of _(1)(I and A() are given as follows. These relations are obtained

from more general results given in Theorem 4.1 of Appendix(1)

JA knI G* dFI + 1/2. 1 = 1.2.....k. (2.1)

k(3n-l) I G~dF. - 2k(2n-1) I Fi * l I * k2 nJ I *2 dFi

ki IHdFi - k2 n( G*dF1) 2 - (n-1) I(I FMdF1)2 - 1/12.

Mal I Jkn(2 - f FjdF I) I G*dF +kn(2 - I FidFj) I G*dFi

n n f dF1 f FadFJ 2kn F a*dF 1 - 2knGd F1 G*dFj81

f I FidFj FjF fI j -dF. +f 1 i J.

(2.2)

where

e(x) = (1/k) Fj(x). (2.3)

H*(x) = (1/k) it F J2 (x). (2.4)

5N w u ~ ~ q ~,~ VV ~ %..%'%~ 9 V~V(2.4).

(2)In case of vector rank R , the moments results are given In Nhtsul (11]

(2) (2)from which we obtain mn vector .-Jp varlance-covariance matrix A 0 of

statistic 1(2) as follow;

n[2f GdFI - 2kf Fi G'dF k2 f *2 dF1

(2) - kJ I *dFi -k ( GdFi ) 2 - 1/12]. I a J.

n(k(2 - I FjdFi) J G*dFj + k(2- f FIdFj) f G dFi"

I f FmdFi f FadF -2 A F. GtdF i -2k f Fi GdFj

f FdFj f FjdF i +*f Fi2dFj +1 FJ2 dF1 - 1]. 1 f . )(2.6)

2.2 ftuwtion

Ve assume the following relation to hold between the gaps of parameters:

1P (Gi. ej) C Cp + KPii n-1/2 * o(n-1/2). P= 1.2. (2.7)

where c. is given by (1.9); for each pair (i.), p ij depends on G i and 9j

and Is inweasing in i hen Gj is fixed, and decreasing in j when Si is

fixed; also, K . ij = C when i = 8j.

Then putting

I -j / ( f Fj(x)dFi(x) - I Fi(x)dFj(x)). (2.8)

we obtain the folloving lema.

Lem 2. 1

For P (# 0G j) (A - 1.2) given by (2.7). we have the following:

10 = 1 i o(1). (2.9)

where

4.-.

. ,,.; ' . ,'.','..' '..%'..,-.% . %°. • . . - , ,, . . . . . , . o.. .. . .. • . . . 4, .. • , • .9.:

O I lij I f 2 (x)dx. when P = 1.j LiL J (2.10)

K2ij Ixf2 (x)dx, when P = 2.

lij 2 L,29...,k; 1 0 j.

Example:

When F(x) is N(0,1), TP (0.0 G ) given by (2.8), we get

Ill i (1/2-/-7)K1ij + o(1) (2.11)

and

121j (1/7) 2i o + 0(1). (2.12)

2.3 Asymptotic Distribution

Let us defineV (1/5 ( a - (a)). .. (2.13)

~k I ,a=..(.3

Then

W(a) = (1/.(i) a H~' ) a = 1.2, (2.14)

where (a) = (l (a). V2 (a), .... Vk(a)). A (-E(k-l), 1(k)) (k-)xk and

is a unit mtrix of order k-I, J(k) = (1, 1, ... , 1)' kxl ) has(a) (a) :

mean vector _6 . variance-covariance matrix Z such that

(a) (a), = (1/.-) AP,# (2.15)

and

(a) (a)= (1/n) A A '. (2.16)

Elements of and .Zp are given by

(a) (a) (a)1 i (1/Jn )(Ppk " lp i) i 1.2....,k-1. (2.17)

(a) (a) (a) (a) (a)CTij = (l/n)(AAIj - A k - AAkJ Afikk ). ij 1.2....k-

(2.18)

7

(a) (a)where A and Ap Ij are given by (2.1) through (2.6).

Now, under the assumption (2.1). using Lem 2.1, we have for P = 1.2 and

a = 1.2.(a) j

17, = (1A) n IjflFjdFk - n I itlFJdFi)

(a)-. (a 1(2.19)

j4 J1 A PA ij-01

as n -# 0o where KP ii is given by (2.10). Also, under (2.7), we have

-k/12 for i 0 'ft

m (k2 -k)/12 for I = J2 -(k 1)/12 for l j

A21J (k2 - 1)/12 for i =J.

Consequently..

(a) " Va for 1 0 j-. i (2.20) ;

I 2v0 for i =

wherek2/12 when a = I

v a = (2.21)k(k + 1)/12 when a a 2.

Thus by applying the central limit theorem, ye have the following asymptotic

distribution of W (a):

-(a) (,), 1,2 (2.22)

(a) (a) (a) (a)where I ( P2 2 . (k-l)) with elements given by (2.19) '

and(a)i= va(Elkl) k (2.23)

where G(k-l) l_(k-l)J (k-l)" 8

. V .N %ii t %- '.".J% %n.- ."." .. ,., 'u%' .. %%, %--- %.%'. % %. %. %., '.-.- '% .% . """%

3. PCS AND LFC(a)

Using the asymptotic distribution of _W. (a.P = 1.2) given by (2.22). the

P( for the rule R(a.,0m) (a,P.m = 1.2) is given by

P(CSIR(a ,e )) Pr(W13 " - (.) 4 -1))

SPr(1JA3 (5 ) (3.1)Pr Ap i.P + S(P.2)J-(k-1) ) l/ v a 13.1

where

.d / when 1 (3.2)0 when a=2

(a) (a) (a) )(W P (Up - a ---. a (3.3)

(a) _Up g(k-l). Elk-l) + G(k-l)). 34),

and an bn means that lIrn an/bn = I as n -, oo.

For the subset selection approach (a = 1). since

K1k J K1 3 ' 0 ,r

anidKpkj 0

for large n, we have

(1) -1_7 _3 a 0k1.

Also, for indifference zone approach (a 2). with the specification

i)p 0( ek.O e 'tC 13 + 813.

we have

(2) f (k I 2 (x)dx) ,fk 81' ,(lk) when A1(k Ixf2 (x)dx) /n- ( 82'/(1 + 82*)) J(k-l) when 13 2.

Thus we have the following

9

- .' . . , . .;,- ". - . . . . - . . Q . .. .. . , - . . , . . . . - .- .

, 'lam 3. 1

Under the assumption of order restriction (2.7) and for large n, the (asymp-

totic) LFC of the PCS for rules R(a,P.1) (a,0- 1,2) are given by

pj 0 j = 0 i,j = 1,2,...,k; aP= 1,2 (3.5)

and for rules R(at.0 .2) (a. A= 1,2) are given by

K pji = Cp. i.J = 1,2,....k-1, j;

Kki= c *P* i = 1,2.....k-1; a.P= 1,2. (3.6)

Under the asymptotic LFCPCCSIR~a,#,ea) Z Pr(U a)i ((,) (,)/ )_~~) (3.7)

_ S(a ) A - ) + S ( R ) a

where va Is given by (2.21). ,a) is given by (3.2) and 7 (P a) is defined

by f (All) 0 for A 1,2 1 (3)

(kI f 2 lx)dx) /inil for A = 1 (3.8)*51

(k I xf 2 (x)dx) /'n( 82*/(1 + S2)) for P = 2.

The expression in (3.7) can be rewritten as

P(CSIR(a. ,a) ;w ()k-l(x _ ( (p 3 ) + 8(P a))/va)d(x) (3.9)

for a,P, a = 1,2, where )(x) is the standard norm l c.d.f.

Determination of the dp Values

By Theorem 3.1. the df values can be asymptotically expressed as

dp(n) = V2-d f * o(n/ 2), aP = 1.2 (3.10)

where d is the solution of the following equation:

Q0d/O -, d/12- ... d/,/2 ) P*, (3.11)

Q Is the joint cdf of a normlly distributioned vector (VI, V2. .... Vk_1) with

E(Vi) = 0. Var(V i) 1 and Cov(Vi.V J) : 1/2. 1 0 j.

10

4,..~. ~ ~'s> ~ *

Determination of the inimum Common Sample Size

In the subset selection approach, let S denote the size of the selected

subset and Efi [SIR] the corresponding expected value when subset selection rule

R is applied and Q is the true nature of status. Then

Ea [SIR) = 1 P { ij is selected IR) (3.12)

Having determined dl(n) from (3.10), one may determine the common sample size n

by imposing the additional requirement that

Et [SIR] & 1 ( (3.13)

for some 0 < -e < k-i, where 9 lies In a given proper subset of the parameter

space under study, for example, the subset defined by '[?] = 1[21 =

O N1lI= 0 [k] - F (n) for location parameters case and 0 = 12-(2 ...

a[k-13 = 0 [k]/( 1 + A (n)) for scale parameters, where (n) > 0 and A (n) > O.

Asymptotically Relative Efficiency

Given two selection procedures R1 and R2. the (asymptotic) efficiency of R2

relative to R1 is defined by

Eff(R1 .R2 ) = n1 /n2 (3.14)

where the ni are the sample sizes required to satisfy P(CSIR)LFC = P. i 1.2.

Then, by Theorem 3.1 and (3.10). we have the following result:

Eff(R(1,.,I).R(2,P,.)) = 1, P 1,2,

Eff(R(a.1.1).R(a,2,1)) = 1, a 1,2,

EffR(1.2I.R(2. R 2)) = k/(k * 1). P = 1.2.

Eff (R(a .1.2).R (a .2.2))

(81*82*/(1 * 82*))2 (1 xf 2 (x)dx/J f2 (x)dx)2 , a = 1,2. (3.15)

11b

~~e

4. &YDlIX

Let k population Ill. 112- .'. "k be given. w !1! has the associated

coatime- distribution P (x) (s = L.2,....k). Take as observations Xal' .2""

Xn 8 from population 7rs (s = 1.2.....k) and consider the combined (Vilcoxon

type) rank sj of a.,j as stated in Section 1. Then the mans, variances and co-

variances of the ranks Rsj are given in the following.

Theore 4.1

E (Ki) a N 1GdFs 1 +/2 (4.1)

V (Ij) 2N 1 GdF s - 2 I FsGdFs , N2 I G2 da

- NI HdF - N2(1 GdFs)2 - 1/12 (4.2)

Cov(aik.t s) 311 GdFs- 4111 FsGd s a I Fadls)2 - 1/12 (4.3)

Cov(tsllj) N(2 - FtdFs) I GdFt + N(2 - I Fsdt) / GdFs

" n1 ' FadE I Fadft - 2 FtGdFS - 23HI FGdFt

*J FsdFt I FtdFs * I Fs2dFt . I Ft2cF - (4.4)

where s,t = 1.2.....k, s o t; Li 2 192.....n 5, i 0 J; ' = 1.2,....n t and

11 a no (4.5) "

G(x) = (1/1N) ( n Fm x) (4.6)

N (x) " (1/) 1 % F. 2 (x) (4.7)

Proof

Ve sketch the proofs for (4.1) and (4.3) above. The reminlng results are

obtained similarly.

12

Pr(Ri1l = s) Pr(a, of X's. a2 of "i's, ... 0 ak of Xk's A X I(nl-a 1 -1) of Xl's. (n2 -82 ) of X2 's. .... (nk-ak) of Xk's) (4.8)

where a (i = 1.2,...,k) is an integer such that

0 A a n1 - 1. 0 &ai & ni (I = 2.3.....k) (4.9)

Sa. s- (4.10)

and "ai of Xj's". "(ni-a i) of Xi's" should be read as "ai variables out of (Xii.

X12. .... Xini) and remaining (ni-a i) variables", and so forth. Further. sum-

mation I is taken over all k-tuples (al1.2,...Oak) of integers which satisfy the

relations (4.9) and (4.10). From (4.8). we have

E(Rll1) = )r .l . .. (n lF2... F

s=A a 1 /a 2) kak

x (1-F1) (nl-al-1) (1-F2 ) (n2-a2)... (1-Fk) (nk-ak)dF (4.11)

By changing the order of summtion, we haveal a2 ak- 1 -,

1R) 2 szi k-1

x (1-F1) (1-F2 ) (n2-a2)... (1-Fk. 1) (nk-n-%-l)(Fk + kl a i + 1) 1

where the sumiation it is taken over all (k-1)-tuples (ala 2 , .. ak 1 ) of

integers which satisfy the relation (4.9). Adding In turn over ak_1. ak-2...

a1 , we obtain the result for E(RII).

Covar lance :

For s < t, we havePr(RI1 =s. t) = [ Pr(a1 of Xl 's. a2 of X2 's, ... , k of Xk's

6 A b, of Xl's, b2 of X2's, .... b of Xk's

X21 S cl of Xl's, c2 of X2 s, ... ck of Xk's) (4.12)

where a1. bi . c (I = 1.2.....k) are integers such that

13

ai *b i * c1 VaA i =12.,...,k (4.13)j a- - 1. lbj a-t-sa- L, jilj z-,- tll (4.14)

and 14 a a, - I for I a 1.2. V1 a ni for I a 3.4.....k.

Summtion I Is taken over all tuples (a1, ... 9 ak. bI. .... bk, c , ... cc k)

which satisfy the relations (4.13) and (4.14). Then

II 1 t t Pr(RIl 2 s. RI2 = t)S<t

= I t 1 P )(xy) dl(x)F 2 (y) (4.15)

where

P( (y) A F11(x) (FI(y)Fi(x))bi (lbFi(,))c Ia.

By chansing the order of summation, first for a then for t, we have

11.I I it ~C)I P1 (x~y)cF,(x)dF2(,Y)

where k-1 1 aj)* 2 k- a1. ) ( 1bj)-l fI*P 1 aj. 71 z-1 bi + ( 1 :-Ci •u al "l J 3

anda I = nk(nk -1) Fk(x)Fk(y) + 3nk Fk(x) nk Fk(y) * 2

P = nk Fk(x) + nk Fk(Y) + 3

7 1 m nk Fk(x) + 1.

Sumation i Is taken over all tuples (a. ... 9 sk_1 . bI , ... , bk. I , cI .

ck_1) which satisfies the condition (4.13). By carrying out the addition in turn

for a set (as, bi . c). I z k-l.k-2...... we have a reduced form of I. By

proceeding on similar steps for E a t Pr(R11 a s. R21 * t)- we obtains~t

Cov(1 1 ,*11).

For rank suma

T,- ~ s l .... ,k (4.15).=1

14

ve have

E(Ts) = ns E(Rs). s - 1,2....,k (4.17)

Cov(Ts Tt) a ns nt Cov(RsjRtj,). st = 1,2,...,k, s s k, (4.18)

and

j2I f i 0j

=ns(3ns-1) 1 GdFs -2Nns(2ns-1) I FsGdFs + N2ns I G2dFs

-nsJ HdFs - N2ns(f GdFs) 2 - ns(ns-1) lno( FdFs)2 -ns 212(4.19)

Especially, If Fi(x) = F(x) for all 1, then we have

E(Ts) = ns(+l)/2, (4.20)

V(Ts ) a ns(-n s )(N+1)/12, (4.21)

Cov(Ts Tt) Z-nsnt(l+l)/12. (4.22)

Also for k = 2, we have the following:

E(T1) = n1(nil)/2 * n nj I FjdFl , ij = 1,2; J 0 1 (4.23)

V(Ti) = ninj(2ni-1) I FjdF1 + ninj(nj-1) I FJ2 dFi

+ nlnj(ni-1) I F12 dFj - ninj(ninj-1)(I FjdFi) 2 - ninj(ni-l)

iLj = 1,2; i 0 j, (4.24)

Cov(T1.T2) = nln 2[n1 I FldF2 * n2 I F2dF1 - (nl~n2-1) f FldF2 I F2dF1

- (nl-1) I F12 dF2 - (n2-1) I F2

2 dF1 - 1]. (4.25)

Finally, se state an order relation for the expected rank sum. Let (F9 (x))

be a family of distributions stochastically increasing In 9. Then we have the

following.

Theorem 4.2

E(RS) a E(Rt) if and only if Ps A Ft where sot 1.2....,k.

15

% %

' ~ ~ ~ ~ ~ ~ ~ ~ ~ ' 11!Z % . *.~ .- -~.-

[1] Alan. K. and Thompson, J. 1. (1971). A selection procedure based on ranks,Ann. Inst. Statist. Math.. 23. 253-252.

[2] Bartlett. N. S. and Govindarajulu. Z. (1968). Some distribution-freestatistics and their application to the selection problem. An. Inst.Statist. Math., 20. 79-97.

[3] Bechhofer. i. E. (1954). A single sample mltiple decision procedure forranking means of normal populations vith know variances. Ann. Mth.Statist.. 25. 16-39.

(4] Gupta. S. S. and McDonald, G. C. (1970). On some classes of selection pro-cedures based on ranks. Nonparametric Techniques in Statistical Inference(Ed. M. L. Puri), Cambridge Univ. Press. London, pp. 491-514.

[51 Gupta S. S. and Panchapakesan. S. (1979). Multiple Decision Procedures:Theory and Methodology of Selecting and Ranking Populations. John Viley& Sons.

[6] Gupta, S. S. and McDonald. G. C. (1982). Nonparametric procedures inmltiple decisions (ranking and selection procedures). ColloquiaMathematics Socletatis Jinos Bolyal, 32. Nonparametric StatisticalInference (B. V. Gnedenko. N. L. Purl, and I. Vince, eds.), Vol. 1.North-Holland Publishing Company. pp. 361-389.

[7] Gupta. S. S. and Panchapekesan, S. (1985). Subset selection procedures:Reviev and assessment. Amer. J. of Math. Manasement Sciences, 5. 235-311.

[81 Lee, Y. J. and Dudevicz, E. J. (1979). On the Least Favorable Configurationof a selection procedure based on rank sun: Counter example to a postulateof Matsui, J. Japan Statist. Soc., 9. 65-69.

[9] Lehmann, E. L. (1963). A class of selection procedures based on ranks,Math. Ann.. 150, 268-275.

[10] Mtaui, T. (1974). Asymptotic behavior of a selection procedures based onrank sums. J. Japan Statlst. Soc.. 4. 57-64.

[11] Mataui, T. (1985). Momenta of rank vector vith application to selectionand ranking. J. Jaan Statist. So., 15, 17-25.

(12] McDonald. G. C. (1972). Soe multiple comparison selection proceduresbased on ranks. Sankhya, Ser. A. 34. 53-64.

16

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[13] NcDonald. G. C. (1973). The distribution of some rank statistics withapplications in block design selection problem. Sankhya. Ser. A. 35. 187-204.

[15] Purl. P. S. and Purl, N. L. (1968). Selection procedures based on ranks:Scale parameter case. Sakhya, Ser. A. 30, 291-302.

[16] Puri, N. L. and Purl, P. S. (1969). Multiple decision procedures based onranks for certain problems in analysis of variance. Ann. Math. Statist..40. 619-632.

[171 Rizvi. N. H. and Voodvorth, G. G. (1970). On selection procedure based onranks: Counter examples concerning the least favorable configurations,

Ann. Math. Statist., 41, 1942-1951.

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III. SUPPLEMENTARY toCTES 4

ID. KEY WORDS (Contintio an reers@ e iIt neesar and identify b.. block 10..mer)

Selection Procedures, Combined Ranks, Least Favorable Configuration, Momentsbf Ranks, Wilcoxon Type, Friedman Type

20. AISTRACT (Contil... ant reevee side it necessary and Identify by block vniewborl

We consider two types of statistics based on the sums of combined(Wilcoxon type) ranks and vector (Friedman type) ranks. Underlying populationsare supposed to belong to the location or scale parameter family of distributions.

Two approaches - subset selection and indifference zone -of ranking andselection procedures based on these statistics are considered in an asymptotic

framework for selecting the population with the largest parameter value. The%

least favorable configurations of parameters are discussed by conipi.ting the%

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exact moments of these statistics and introducing an assumption of order relationbetween the gaps of parameters.

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