NONPARAMETRIC TESTS FOR INTERC~GEABILITY

UND~R COMPETING RISKS

By

Pranab Kumar Sen

Department of BiostatisticsUniversity of North Carolina at Chapel Hill

Institute of Statistics Mimeo Series No. 905

JANUARY 1974

NONPARAMETRIC TESTS FOR INTERCHANGEABILITY UNDER COMPETING RISKS*

By PRANAB KUMAR SEN

University of North Carolina, Chapel Hill

Abstract

Some nonparametric tests for the hypothesis of interchangeability of the

elements of a (stochastic) 2-vector under competing risks model are proposed

and studied here. Both fixed sample and sequential procedures are studied.

The case of progressively censored nonparametric procedures is also presented.

Along with some martingale theorems on allied rank statistics, their weak con-

vergence results are considered and incorporated in the study of the asymptotic

properties of the tests. The choice of locally optimal score function is also

considered.

AMS 1970 classification Nos: 62G10, 62G20 & 60B10.

Key words and phrases: Asymptotically optimal score function, Bahadur

efficiency, hazard rate, interchangeability, invariance principles, (joint)

survival function, nonparametric tests, progressive censoring and sequential

tests.

*Work supported by the Aerospace Research Laboratories, U.S. Air Force SystemsCommand, Contract No F 33615-C-71-1927. Reproduction in whole or in partpermitted for any purpose of the U.S. Government.

~. For a two-component system, let F(x,y) be the joint distri­

bution function (df) of the survival times X and Y of the two components. We

desire to test the null hypothesis that X and Yare interchangeable, 1. e. ,

(1.1) F(x,y) = F(y,x) for all (x,y)~E2,

where Ek , k>l, stands for the k-dimensional Euclidean space. Nonparametric

tests for (1.1) are due to Sen (1967), Bell and Smith (1969), and others. In

competing risks problems, instead of (X,Y), the observable random vector is

(Z,Q), where

(1.2) Z = min(X,Y) and Q=l, 0 or -1 according as Z=X, Z=X=Y and Z=Y.

For an exposition of joint survival functions under competing risks, we may

refer to Thompson et a1. (1972, 1973) where other references are cited. Thus,

based on a set of observable random vectors (Zi,Qi)' l<i~n, our problem is to

test for (1.1) against suitable alternatives. Nonparametric tests for this

problem are proposed and studied here.

Three different types of tests are considered: (i) the conventional fixed

sample size procedure based on all the n observations through a single statistic,

(ii) the first sequential procedure based on the observations when the Z.care~

observable sequentially, and (iii) the second sequential procedure suitable

under progressive censoring. The first sequential procedure is suitable when

the observations are not available at the same time, so that if the null

hypothesis (1.1) may be rejected based on fewer than n observations, there is

a reduction of the total time to perform the test. In the context of life-

testing problems, when n independent systems are subject simultaneously to a

continuous time-observation process and the (Z.,Q.) are observable only at the~ ~

expiry of the lives of these systems, one may naturally be interested in

2

monitoring the experiment with the objective of rejecting the null hypothesis

with the minimum sacrifice of the lives of the units, that is, stopping the

experiment at a time point where, for the first time, the accumulated evidence

leads to the rejection of H. Unlike the other case, here the ordered randomo

variables corresponding to Zl"",Zn are observed sequentially, and the scheme

is known as a progressively censored scheme. Our second sequential procedure

applies to this situation. Thus, for both the sequential procedures, the stopping

times are random variables, and the procedures may lead to reduction of time and

cost of experimentation. We shall discuss these in greater detail in section 2.

The test procedures along with the preliminary notions are introduced in

section 2. Some martingale theorems, invariance principles and certain basic

invariance structures for the allied rank statistics are studied in section 3.

Section 4 is devoted to the study of the properties of the fixed-sample and first 4Itsequential tests based on appropriate rank statistics. Their asymptotic relative

efficiency (ARE) results are also considered. Parallel res~lts for the second

I sequential procedure are presented in section 5. The last section is concerned

with the choice of optimal scores.

notions and the roosed tests. Let {(Z.,Q.), i>l} be a sequence~ ~ -

of independent and identically distributed random vectors (iidrv), where the

(Z.,Q.) correspond to (Xi,Y.) as in (1.2). We assume that F(x,y) possesses a~ ~ ~

density function f(x,y), V(x,y)EE 2, so that (i) the density function (say, g(z»

of Z. exists, and (ii) P{Xi=Yi } = p{Q.=O} = 0, Vi>l. Let G(z) be the df of Z.,~ ~ - ~

so that G(z) is absolutely continuous in ZEE. Hence, ties among Zl'.'.'Zn can

be neglected with probablity 1.

Let c(u)=l or 0 according as u is > or <0, and ,let Rni = Lj:l c(Zi-Zj) be

the rank of Z. among Zl""'Z , for l<i<n. Thus, R =(R 1, ••• ,R ) is some~ n -n n nn

3

permutation of (l, .•. ,n). For every n(>l), consider a set of real-valued rank-

scores a (l), ••• ,a (n), defined byn n

(2.1) a (i) = E¢(U .) or ¢(i/(n+l»,n n.~

where U 1 < ••. < U are the ordered random variables of a sample of size nn nn

from the rectangular (0,1) df [so that EU .=i/(n+l), l<i<n], and the score-n~ --

function ¢(u), O<u<l, is assumed to be square-integrable and non-degenerate,

so that

(2.2)1

o < A2 = f ¢2(u)du < 00.

o

Consider first the fixed-sample size test. Define the rank statistics

(2.3) n>l.

As we shall see in section 3 [cf. Lemma 3.4] that under Ho in (1.1), gn=(Ql, ••• ,Qn)

and R are stochastically independent and Q assumes the all possible 2n realiza--n -n

tions, each with the equal probability 2-n • Thus, under H , E(T )=0 ando n

-1On the other hand, when H does not hold, n To n

estimates a quantity which may be positive or negative depending on the df F(x,y).

Thus, for a one-sided test, we may consider the critical region specified by

(2.4)

while for the two-sided test, our critical region is given by

(2.5) n~IT IfA > c(2), where P{n-~IT IfA > C(2)} = a.n - n,a n n - n,a

In section 4, we shall see that the tests sketched above are genuinely dis-

tribution-free, so that C(i)n,a'

a (not on F), and there exist

i=1,2, depend only on the level of significance

suitable c~i), i=1,2, such that

(2.6) limn400 C~~~ = c~i), i=1,2, for every O<a<l.

4

Let us next consider the first sequential test. Here the i.i.d.r.v.

(Zi,Qi)' i~l, are observable sequentially, so that it may be advisable to stop

at an intermediate stage i.e., when (Z.,Q.), l<i<k, are observed for some k<n,, ~ ~

provided the statistical evidence up to that stage provocates the rejection of

Ho • For every k: l~k<n, define Tk by (2.3), and conventionally let To=O. Let

then

(2.7)

so that corresponding to (2.4) and (2.5), we consider the critical regions:

(2.8)

(2.9)

M+ > M+ where P{M+ > M+ IH} = a,n - n,a' n - n,a 0

M > M where P{M > M IH} = a.n - n,a' n - n,a 0

Operationally, the test procedure consists in observing sequentially the Tk , k~l, ~

~ -1 ..b.< -11 I ,'+ •until for the first time for some k=N~n), n A TN(or n 2A TN) exceeds Mn n n,a

(or M ), and rejecting H at that stage with the termination of the experiment.n,a I 0

If no such N(~n) exists, then Ho is accepted when (Zl,Ql), ••• ,(Zn,Qn) are observed.

We shall see in section 4 that the' test procedure is distribution-free, so that

M+ or M does not depend on the underlying F, and further, there existsn,CI. n,CI.

suitable constants M: and Ma , such that

(2.10) + +lim ~M = M and lim -+ooM = M •n~ n,CI. CI. n n,CI. a

Finally, let us consider the progressively censored rank test. In this

case, the experimentation starts with the continuous observation on n units and

their values are recorded as they are observed sequentially. Thus, here the

order statistics Z 1 < .•• < Z (corresponding to Zl' ••• 'Z ) are observed inn, - - n,n n

a sequence; by virtue of the assumed continuity of G, ties among the Z i can ben,

neglected with probability one. We may note that

(2.11) Z . = Zs .' l~i~n,n,1., n~

(2.12)

5

where 8 =(8 1, ••• ,8 ) is some permutation of (l, ••• ,n). In view of the fact-n n nn

that Z. = Z R ,l<i<n, we term 8 as the vector of anti-ranks. Also, we denoteJ. n - - -n, nithe Qj corresponding to Z8 . by Q8 . = Q(n,8ni), for i=l, .•• ,n. Then, we observe

nJ. . nJ.that at the kth stage when Z l""'Z k have been observed, we are providedn, n,

with Q(n,8nl), ..• ,Q(n,8nk), for k=l, ••• ,n. We denote by

Tnk = Li : l Q(n,8ni)an (i), l<k<n.

Note that, by definition,

(2.13) T = L.nl R(n,8 .)a (i) = L.n

l Qia (R .) = T , n>l.nn J.= nJ. n J.= n nJ. n

Conventionally, we let T =0 and T =0, Vn>O, and defineo no -

D+ = { maxk

{ max IT k/}I (n~A ).(2.14) T k}/(n 2A ) and D =n O<k<n n n n O<k<n n n

For an one-sided + and reject H whentest, we use Dn 0

(2.15) D+ > D+ where P{D+ > D+ IH} = 01.,n - n,OI. n - n,OI. 0

and for a two-sided test, we use D and reject H whenn 0

(2.16) D > D where P{D > D IH} = 01..n - n,OI. n - n,OI. 0

Operationally, the test procedure consists in continuing the experiment so

4 -1 4 -11 I +long as nAT k (or nAT k ), l<k<n, continue to lie below D (or D )n n n n -- n,OI. n,OI. '

and if N«n) is the smallest pQsitive integer for which n4 A-l T N is > D+ (or- n n - n,OI.

n~A-lIT NI is > D ), the experimentation is terminated along with then n - n,OI.

rejection of Ho ' If no such N(~n) exists, Ho is accepted. In section 5, we

shall see that the tests based on D+ and D are genuinely distribution-free, andn n

+there exist suitable constants DOl. and DOl.' such that

(2.17) lim D+ = D+ and lim D = D , V 0<01.<1.n-?OO n,OI. 01. n-?OO n,OI. 01.

For the study of the various properties of these tests, we require to study

first some basic properties of {Tn' n~l} and {Tnk

, l<k~n}. This has been

6

accomplished in section 3.

given by

The density function g(z) of Zi is

00 00

(3.1) g(z) = J f(z,y)dy + J f(x,z)dx, -oo<z<oo.Z Z

Let n(z) = p{z=xlz=z} = p{Q=lIZ=z}, -00< z<oo , so that00

(3.2) n(z) = [J f(z,y)dy]/g(z), O<n(z)~l, -oo<z<oo.Z

For every l<i~n, let

(3.3)

(3.4)

00

n-1 J i-1. n-in(i,n) = n(i-1) n(z)[Gz)] [l-G(z)] dG(z), n*(i,n) =-00

- -lL n (') *(' ) d * - L n~ - n '1 a ~ n 1,n an ~ - k l~k·n ~= n n =

2n(i,n)-1;

Note that under H in (1.1), n(z)~ for all -oo<z<oo, so thato

(3.5) n(i,n) = ~, n*(i,n) = 0, l<i<n and ~ =~*=O, V n>l.-- n n

For every n>l, let B be the a-field generated by (Q ,R ) where Q and Raren ~n -n ~-n

defined in section 2. Note that B is t in n~l). Then, we have the following:n

Theorem 3.1. For an(i)=E¢(Uni), l~i~n and ¢ integrable inside [0,1], {Tn-~~,Bn; n>l}

is a martingale.

Proof.

(3.6)

By (2.3), (3.3) and (3.4), E(T1-~1)=0, while for n>2,

E(T -~*IB 1) = [L~-llE{Q,a (R ,)IB 1}-~* 1]n n n- 1= 1 n n~ n- n-

+ [E{Q a (R )IB 1}-~].n n nn n- n

Now for 1<i<n-1, given B l' R , can be either R l' or (R 1,+1) with respective- - n- n1 n- ~ n- ~

-1 -1conditional probabilities 1-n R 1i and n R 1" and Q, is fixed, so thatn- n- ~ 1

(3.7) I { -1 -1E{Q,a (R ,) B 1} = Q, (l-n R l,.)a (R 1i)+n R l,a (R 1,+1)}1 n n~ n- ~ n- 1 n n- n- 1 n n- 1

= Q,a l(R 1')' 1<i<n-1,1 n- n- ~ --

7

where the last step follows from the well-known and easily verifiable identity:

(3.8) n-l[(n-i)E¢(Un .) + iE¢(Un .+l )] = E¢(Un_li), l<i<n-l.1. 1.

Thus, from (3.6) and (3.7), we have

(3.9) E{T -]1*IB I} = T 1-]1* 1 + E{O a (R ) IB ·1}-]1 •n n n- n- n- -n n nn n- n

Now, given B l' the possible values of 0 a (R ) are ±a (j), j=l, ••• ,n, andn- -n n nn n

Q a (R ) = a (j) with probability ~(j,n), and -a (j) with probabilityn n nn n n n

!(l-TI(j,n», for j=l, •.• ,n, so thatn

(3.10) E{Q a (R )IB I} = L.nla (j)!{2TI(j,n)-1} = ]1 , by (3.4).n n nn n- J= n n n

Hence, the theorem follows from (3.9) and (3.10). Q.E.D.

Since Q~=l with probability 1, for every i>l, by (2.3) we have1.

(3.11)

= \inl a2(i) + Ll<.4.< E{Q.Q.a (R .)a (R .)},I.. = n _1.;J_n 1. J n n1. n nJ

where by (2.1) and (2.2), as n+oo,

(3.12)1

= I ¢2(u)du.o

We let TI*(z) = {2TI(z)-1}, -oo<z<oo, and for l<k<~n(~l), define

(3.13) TI*(k,q;n) n!=~---:,...--,---'=":".,....,-,..-,--~

(k-l)! (q-k-l)!(n-q)! II k-l q-k-lTI*(u)TI*(v) [G(u)] [G(v)-G(u)]-oo<u<v<oo

[l-G(v)]n-qdG(u)dG(v).

Note that under Ho ' TI*(k,q;n) = 0, Vl~k<~n. Since the Zl"",Zn are iidrv,

by some standard arguments, it follows that for l<i:fj<n,

8

u v }Q.=(-l) , Q.=(-l) ]~ J

P[R .=s, R .=t,n~ nJ

E{Q.Q.a (R .)a (R .)} = . I a (s)a (t) J u=I.o

v=I'O(-l)U+V~ J n n~ n nJ s;'f=l n n )

2 an (k):an (q)rr* (k,q;n).l<k<:~n

Thus, from (3.11) and (3.14) and by Theorem 3.1 (~ ET =~*), we haven n

(3.14)

(3.15) VeT ) = nA2 + 2 2 a (k)a (q)rr*(k,q;n)-(~*)2.n n 1<k<~n n n n

It readily follows from (3.5) and the fact that under H , rr*(k,q;n)=O, thato

(3.16) VeT IH ) = nA2 and n-lV(TIH ) -+ A2 as n~.non n 0

To simplify (3.15) for large n when H is not necessarily true, we assumeo

that the fol,lowing conditions are satisfied:

(I) ~(u) = ~1(u)-~2(u) where ~j(u) is non-decreasing and absolutely con­

tinuous inside [0,1], and

(3.17)

and (II) rr(z) is absolutely continuous in z for all O<G(z)<l.

Let us then define1 00

(3.18) (j2 = (j2 (F) = J ~2(u)du - ( J rr*(z)~(G(z»dG(z»2 +0 -00

2[ JJ rr*(u)rr*(v)[G(u){l-G(v)}~I(G(u»~'(G(v»+~(G(u»{l-G(v)}~'(G(v»-oo<u<v<oo

- G(u)~'(G(u»~(G(v»]dG(u)dG(v)].

Note that !rr*(z)I .~ 1, ~ -oo<z<oo, so that some standard computations yield that

a 2(F)<00 for every F. Then, we have the following.

Theorem 3.2. Under (2.1), (3.17) and conditions I and II,

(3.19)

Proof.

that

(3.20)

9

-1 2.. n V(T ) -+- q as n~.n

By virtue of (3.17), we obtain, on proceeding as in Hoeffding (1973),

Consequently, if we prove the theorem for a (i) = E~(U .), l<i<n, the resultn 'r n~ --

applies as well to the other case of an(i) = <p(i/(n+1», l<i<n.

and for n>l,

We let T =0,o

(3.21) L = T -T 1-~' 02 = E(L2 IB 1)' n>l.n n n- n 11 n n- -

Then, by Theorem 3.1, we have n-1V(T ) = n-1~.n1E[q2], so to prove (3.19), itn L~= n

suffices to show that as n~,

(3.22)

By (2.3), (3.21) and a few steps we obtain that

2 2 I 2 In- 1 .= EaR B - + EaR -a R 2 Bqn [n( nn) n-1] ~n i=l {[ n( ni) n-1( n-1i)] In-I}

(3.23) + I Q.Q.E{[a (R .)-a l(R l·)][a (R .)-a l(R l,)]IB I} +1~i~j~n-1 ~ J n n~ n- n- ~ n nJ n- n- J n-

In- 1 I2 , 1Q,E{Q a (R )[a (R i)-a l(R 1i)]B I}·

~= ~ n n nn n n n- n- . n-

Now, as in the proof of Theorem 3.1, we have

(3.24)1

= f <P2 (u)du as n~,

o

(3.25)

(3.26)

E{[a (R .}-a l(R l,)]2I B l} =n n~ n- n- ~ n-

-2 2[n R 1.(n-R 1.)][a (R 1.+1)-a (R 1')]' 1<i~n-1,n- ~ n- ~ n n- ~ n n- ~

E{[a (R ,)-a l(R l·)][a (R ,)-a l(R l,)]IB l' i~j}n n~ n- n- ~ n nJ n- n- J n-

-2= n WI (n-w2) [an (w1+1)-an (wl )] [an (w2+1)-an (w2)] ,

10

(3.27) E{Q a (R )fa (R i)-a l(R I')] 18 }n n nn n n n- n- 1. n

= ~,n1a (j )1".* (j ,n){[a (R I'+c (R I' -j ) )-a 1 (R I' ) ]}LJ= n n n n- 1. n- 1. n- n- 1.

. . -2 IRn-1i= [a (R l,+l)-a (R 1i)]{n (n-R I') '1 TI*(j,n)a (j)n n- 1. n n- n- 1. J= n

- n-2R 1'~.~ R +1 TI*(j,n)a (j)}.n- l. LJ=, nn-11.

Also, note that for 1<i~j~n-1,

(3.28)

(3.29)

E(Q,IR 1) = 2TI(R l"n-1)-1 = TI*(R l"n-1),1. ~- n- 1. n- 1.

E(Q,Q,IR 1) = TI*(R l"R l,;n-I), if R I' < R I'1. J -n- n- 1. n- J n- 1. n- J

= TI*(R 1j,R l,;n-1), if R Ii > R I'·n- n- 1. n- n- J

Thus, writing E(q2) = E{E(q2IR I)}' and using (3.23) through (3.29) thatn n -n-

(3.30)

2 I TI*(i,j,n-I)n-2i(n-j)[a (i+1)-a (i)][a (j+I)-a (j)] +1<i<j~n-1 n n n n

2 L~:iTI*(i,n-1)[an(i+1)-an(i)]{n-2(n-i)Ij:1TI*(j,n)an(j) -

-2 . ~n *(' ) (') }n l.Lj=i+1TI J ,nan J •

Note that TI*(z) = 2TI(z)-I is a bounded and absolutely continuous function of z,

so that by the well-known bounds for expected order statistics, we have

(3.31)

(3.32)

-1 ~TI*(i,n) = TI*(G (i/(n+1») + o(n ), l~i<n,

-1 -1 ~/-TI*(i,j;n) = TI*(G (i/(n+1»)TI*(G (j/(n+1») + o(n 2), l~i<j<n.

Thus, by (3.4), (3.31) and Hoeffding (1953, 1973), we obtain that00

(3.33) ~ +~(F) = f {2TI(z)-1}¢(G(z»dG(z) as n~.n -00

Also, by the recent results of Hoeffding (1973), the third term on the rhs

11

(right hand side) of (3.30) converges to 0 as n+oo. By (3.32) and some standard

steps, the fourth term on the rhs of (3.30) converges to (as n+oo)

(3.34) 2 JJ n*(x)n*(y)G(x)[l-G(y)]¢'(G(x»¢'(G(y»dG(x)dG(y),_oo<x<y<oo

and similarly, the last term converges (as n+oo) to

(3.35) 2 JJ n*(x)n*(y){¢(G(x»[l-G(y)]¢'(G(y»-G(x)¢'(G(x»¢(G(y»}dG(x)dG(y).-oo<x<y<oo

The proof of (3.22) follows from (3.18), (3.12), (3.28), (3.33), (3.34) and

(3.35). Q.E.D.

Now, by virtue of Theorems 3.1 and 3.2, for every O<s~t~l,

(3.36)

In the sequel, it will be assumed that rr2 is strictly positive, so that

(3.37)

Let 1=[0,1], W (0)=0, n>l, and definen· -

(3.38)

. { k k+lConsider then a stochastic process W= W (t),t£I}, where for - < t < --- ,, n n n- - n

(3.39) k k+l kWn (t) = Wn (n)+(nt-k)[Wn (7)-Wn (n)]' k=O, ••• ,n-l.

Thus, for every n~l), W belongs to the space C[O,l] with which we associate then

uniform topology specified by the metric

(3.40) p(x,y) = sup Ix(t)-y(t)I , x,y£C[O,l].t£I

Finally, let W={W(t),t£I} be a standard Brownian motion on I, so that EW(t)=O

and E[W(s)W(t)]=min(s,t) for every s,t£I.

Theorem 3.3. Under (3.37) and the conditions of Theorem 3.2,

(3.41)V

Wn ~ W, in the uniform topology on e[O,l].

12

Proof. As in Hajek (1968) and Hoeffding (1973), for every n>O, there exists

[under (3.17)] a decomposition

(3.42) ¢ (u) = ¢ (1) (u) + ¢ (2) (u) - ¢ (3) (u), O<u<l,

where ¢(l) is a polynomial, ¢(2) and ¢(3) are non-decreasing, and

(3.43) {rj~2 (I~(j) (u)! {u(l-u)}-<'du } <nr(IHu)! {u(l-u)}-<'du.

Now, in (3.18), on replacing ¢(u) by ¢(j)(u) everywhere and denoting the correspond­

ing quantity by a~, j=1,2,3, it can be shown that (3.43) implies thatJ

(3.44)

where n'(>O) depends on n, and n'~O as n~O. Also, by virtue of (3.20),

l~~nln~{Li~l Qi~(~i) - Li~l Qi¢«k+l)-l~i)}1 = 0(1), for ~(i) defined

by (2.1). Hence, here also, it suffices to work with an(i)=E¢(Uni), l<~n(>l).

Suppose now in (2.3) and (3.4), we replace the score function a (i) byn

a j(i) = E¢(.)(U .), l<i<n, j=1,2,3, and.denote the corresponding quantities byn, J n1 --

T ., ~ . and ~* ., respectively, for j=1,2,3. Similarly, in (3.38)-(3.39), wen,J n,J n,J

replace Tk'~*k and a by Tk "~k* . and a., respectively, and define the resulting,J,J J

process by W . = {W .(t),t£l}, for j=1,2,3. Then, by (3.42), we havenJ nJ

(3.45)

Note that Theorem 3.1 applies to each of {T .-~* .,B ;n>l}, j=1,2,3, and byn,J n,J n -

definition, SUPt llw .(t)l= max, n-~ITk .-~~ .lla., so that by the Kolmogorov-£ nJ O<k<n ,J ,J J

inequality for martingales, we have

(3.46) p{suplW .(t) I ~ K} = p{ max ITk .-~~ .1 ~ Kvna.}t£l nJ O<k<n ,J,J J

< (nK2 )-lE[T j-~ .]2/a~ ~ K-2 , as n-+<x>, j=1,2,3.n, n,J J

13

by Theorem 3.2. By virtue of (3.44), (3.45) and (3.46), for every £>0 and E'>O,

there exists an n>o, such that under (3.17) and (3.42),

(3.47) p{suplw (t)-(rrZ/cr)w 2(t)+(rr3/rr)W 3(t)I>E'}<E.tEl n n n

Consequently, by (3.44) and (3.47), it suffices to prove that as n~,

(3.48)V

Wn1 + W, in the uniform topology on C[O,l],

and for this purpose, we use a functional central limit theorem for martingales

[cf. Theorem 3 of Brown (1971)] according to which it suffices to show that as

n~, for every E>O,

(3.49)

(3.50)

-II n 2 I In '·lE{L, II ( L. 1 > Err1Vn)} + 0,~= ~, ~,-

~ n 2 2 P(£'-1 q, 1)/(nrr1) + 1,~- ~,

where Ln ,l and qn,l are defined by (3.21) for ~=~(1) and I(A) stands for the

(r) r/ rindicator function of a set A. Let ~(l)(u) = (d du )~(l)(u), r=0,1,2. Since

~(1) is a polynomial and is absolutely continuous, we have

(3.51) sup 1~«lr»(t)1 = K «00), for r=0,1,2.O<t<l r

1< IT1 1/ + la1 1(1)/ = 2/1 ~(l)(u)dul < 00, and for n>2,

, , 0

(3.52) IL 11 ~ I~=11Ia l(R ,) - a -1 l(R -I') I + la l(R ) 1 + IlJ li-n, ~- n, n~ n, n ~ n, nn n,

Note that R , is either R Ii or R 1,+1, so that on using (3.8) and (3.51),n~ n- n- ~

(3.53) la l(R .)-a 11(R '1,)1 < la l(R l,+l)-a l(R 1,)1n, n~ n-, n- ~ - n, n- ~ n, n- ~

~ max la 1(k+1)-a 1 (k)/ = o(n-1),1<k<n-1 n, n,

as under (3.51), n[a 1(i+1)-a l(i)] = a l(i) is bounded, andn, n, n,

(3.54) Ian,l(i)-<P ~g (i/ (n+l) I -+ 0, as n-+oo, 'V l~i<n.

14

Similarly, la l(R ) I ~ max /a l(k) I = 0(1) and I~ 11 = 0(1). Consequently,n, nn l<k<n n, n,

by (3.52), (3.53) and the above; we have that for every £>0, there exist an

integer n£, such that

(3.55)

On the other hand,

-I, n£ 2 I I Cl _l,n£ 2(3.56) n Li=lE{Li,lI( Li,l > £crlynI ~ n Li=lE(Li,l)

= n-Lv[T 1] ~ cr12(n In) -+ 0 as n-+oo.

n , ££

Hence, (3.49) follows from (3.55) and (3.56).

To prove (3.50), we use (3.23) through (3.27) for <P=<P(1) i.e., an ,l(i),

l<i<n. Writing then w" = mineR l"R I') and w~, = max(R li,R lj)' we have1.J n- 1. n- J 1.J n- n.,.

(3.57)

, -22 L QiQ,n w,,(n-w~,)[a l(R l,+l)-a l(R li)][a l(R lj+1)-a l(R 1j)]1<i<j<n-1 J 1.J 1.J n, n- 1. n, n- n, n- n, n-

- -R

In- 1 -2 n-1i * ' ,+ 2 i 1Q,[a l(R l,+l)-a l(R l,)]{n (n-R 1')" 1 n (J,n)a l(J)= 1. n, n- 1. n, n- 1. n- 1. LJ= n,

-n-2R -I' I~=R +1 n*(j,n)a l(j)}·n 1. J I' n,n- 1.

1The first term on the rhs of (3.57) converges to f <P(1) (u)du as n+oo, and the

00 0second term to ~i(F) = ( f n*(z)<p(1)(G(z»dG(z»2. By (3.53), the third term

-00

goes to 0 as n+oo, while the fourth term can be written as

(3.58)

where the S , and Q(n,S ,) are defined by (2.11) and shortly after that. Noten1. n1.

that as in (3.31),

(3.59) E[Q(n-1,S l,)Q(n-l,S I')] = n*(i;j;n-1), for 1_<i<J'<n-1,n- 1. n- J

15

so that by (3.54), (3.32) and (3.59), the expected value of (3.58) converges

(as n-+oo) to

(3.60) 2 JJ ~*(x)~*(y)G(x)Il-G(Y)]d¢(l)(G(x))d¢(l)(G(y)).-oo<x<y<oo .

On the other hand, Q(n-l,Sn_li)' l<i<n-l, are interchangeable and bounded (by 1)

random variables, so that on evaluating the 4th moment of (3.58), using the

Markov-inequality and the Borel-Cantelli Lemma, it follows that (3.58) converges

almost surely to (3.60). In a similar manner, it follows that the last term on

the rhs of (3.57) converges almost surely (as n-+oo) to

(3.61) 2 JJ ~*(x)~*(Y)I¢(l)(G(x))Il-G(y)]¢~i~(G(Y))-oo<x<y<oo

- G(x)¢~i~(G(x))¢(l)(G(Y))]dG(X)dG(Y).

Thus, q~,l -+ of almost surely as n-+oo, and this implies (3.50). Q.E.D.

Remark. On using (3.42)-(3.43) and the recent results of Hoeffding (1973), it

can be shown that under (3.17), (3.37) can be improved to:k

(3.62) In2I~n-~(F)]1 -+ 0 as n-+oo,

Consequently, in (3.38), it isso that { max Ik~(F)-~~l/l:no} -+ 0 as n-+oo.l<k<n

possible t~ replace ~~ by k~(F) for l<k<n.

Let us now consider the situation when Ho holds.• jl . jn

Let J =«-1) , ••• ,(-1) )-n

where j1.' is either 0 or 1, l<i<n, and let J ={j : j.=O,l, l<i<n}. Also, let- - -n _n 1. --

S be the set of all possible n! realizations of S , defined after (2.11).n -n

Finally, let g(~n) = (Q(n,Snl), .•. ,Q(n,Snn))· Then, we have the following.

Lemma 3.4. Under H in (1.1) Q =(Ql, ••• ,Q ) and R =(R 1, •.. ,R ) are stochasticallyo - -~ n -- -n n nn

independent, and for every S £S ,-n n

(3.63) P{Q(S ) = j }- -n -n for every j £J •_n -n

Proof. Now Zl, ••• ,Zn are iidrv, so that gn can have all possible n! permuta-

tions of (l, ••• ,n) with the common probability lin!. On the other hand, if r-n

16

is any permutation of (l, ..• ,n), then

(3.64) p{Q =j , R =r IH }~n ~n ~n ~n 0

n l-ji ji= f ••• f{ IT {g(Zi)[TI(Z.)] [l-TI(z.)] }dz.,

( ) i=l ~ ~ ~ ~~n

where the n fold integration extends over the domain {~Z <••• <Z <oo} andsnl snn

~n is the anti-rank ve~tor corresponding to En' Since, by (3.1) and (3.2), under

H , TI(z)~ for all Z, (3.64) reduces too

(3.65)-n .

2 P{R=r}.~n ~n

Hence, p{Q =j IR =r ,H } = 2-n , V r , and this implies the independence of R~n ~n ~n ~n 0 ~n ~n

and Q •~n

(3.66)

Hence

p{Q = j IH } = 2-n, V j EJ •-n ~n 0 ~n ~n

By virtue of the fact thatS is the anti-rank corresponding to some R , we-n ~n

have p{Q(S )=j IH } = pro =j IR ,H }, and hence, (3.63) follows from ("3.66)·~ ~n ~n 0 ~ ~n -n 0

and the independence of Q and R. Q.E.D.~n -n

(k) (k)Let S =(S l""'S k)' Q(S ) = (Q(n,S l),···,Q(n,S k»' and let B*k be

~n n . n '" n n n n(k) (k)

the a-field generated by (S ,Q(S », when H holds, for k=l, ..• ,n.-n ~ n 0

Lemma 3.5. For every n>l, {Tnk,B~k' l<k~n} is a martingale.

Proof.

(3.67)

By (2.12), for every k>q,

E(T klB* ) = E{L.kl a (i)Q(n,S .)IB* }n nq ~= n n~ nq

= T + L.k

+1 a (i)E[Q(n,S i)IB* }.nq ~=q n n nq

By Lemma 3.4, for every i>q, E[Q(n,S )/B*] = 0, so that by (3.67),ni nq

E(T klB* ) = T for every k>q. Q.E.D.n nq nq

We let T =0, and for l<k<n,no

(3.68) W*(k/n)n

4 -1= nAT k'n n

17

and by linear interpolation between [k/n,(k+l)/n], for k=O,l, .•• ,n-l, we

complete the definition of W*={W*(t),tEI}.n n .

Theorem 3.6. Under (1.1), (2.2) and the condition that

(3.69) max {Ia (k)I/i'n) + 0 as n+oo,l<k<n n

W* g W, in the uniform topology on C[O,l],n

[Note that (3.17) implies (3.69) but the converse is not true.]

Proof. Let ~nk = Tnk-Tnk-l' l<k<n. Then, by (2,12) and Lemmas 3.4 and 3.5, we

have

(3.70) a 2 (k) ,n

Also, by (3.69), I~nkl = lan(k)I ~ max la (i)l=l<i<n n

(3.71) v~ = Lk~l E(~~kIB~k_l) = Lk~l a~(k) = n A~,1

where A2 + A2 = J ¢2(u)du, as n+oo,n 0

o(V ), for all k: l<k<n. Hence, for every E>O,n

(3.72)

The convergence of the finite dimensional distributions of W* to those of Wn

follows directly from (3.71), (3.72) and Theorem 2.1 of Dvoretsky (1972). By

virtue of Lemma 3.5 and the Doob upcrossing inequality for semi-martingales,

the proof of the tightness of w* follows along the lines of Section 6 of Brownn

(1971), and hence, the details are omitted.

under Ho

+44rf.fI/X)rR~mAjfAK!?,,,R/~fI;tR~tflA:KI?:;r!?,,,J?JWrJtf'lJm T M ~mr1 M By virtue of Lemma 3.4,r\., n' n I'r:J'\i"i; n'

in (1.1), Q and R are stochastically independent, and p{Q =j IH }-n -n -n -n 0

-n= 2 for every j EJ •-n -n

Thus, if we let

(4.1) T = \.nl

a (i)U., n>l,n L.J.= n J.

where U., l_<i<n, are iidrv, and P{U.=+l} = ~ i>l, we conclude that T has theJ. J.- , n

(4.2)

18

- -same distribution (under H ) as of T. On the other hand, T involves a linearo n n

combination of iidrv, and hence, its distribution can be traced without much

problem. In fact, if one keeps in mind the classical one-sample problem, then

the corresponding rank order test statistic [viz. Hajek and Sidak (1967, p. 108)]

-has the same distribution (under the null hypothesis of symmetry) as of T •n

Consequently, the available tables for this situation [viz., Owen (1962)] for

various common scores and small sample sizes provide the necessary tables for

-our case too. Since, the distribution of T depends only on a (l), ••• ,a (n),n n n

we conclude that under H in (1.1), T is genuinely distribution-free. On theo n

other hand, by Theorem 3.3, it follows that for every real x,

-k 1 xlim pin 2 Tn/A

n<xIHo} = I!TI J exp{~t2}dt = ~(x),

n~ -J::JO

where ~(x) is the standard normal df. Thus, if ~(Ta) = I-a, O<a<l, we obtain ~

from (2.4), (2.5) and (4.2) that

(4.3) lim C(l)n,a

n~

= Ta and lim C~~~ = Ta/2 , O<a<l.n~

With a view to studying the ARE of the proposed test for various score

functions, we first consider the Bahadur-efficiency of the tests. For this,

we first consider the following.

Lemma 4.1.

(3.33) •

-1Under (3.17), n T + ~(F) a.s., as n~, where ~(F) is defined byn

Proof. Using (3.42), we rewrite T = T 1 + T - T 3' where by the Schwarzn n, n,2 n,

inequality and (3.43), for j=2,3,

(4.4) I -1 I -II n 2 ~ -II n 2 ~n T . < (n . 1Q·) (n i la .(i»n,] - ].=]. = n,]I nk

= [- I a 2 .(i)]2 < ~n',n i=l n,]

for n ~ no(n), where n'>O and n'+O as n+O. Thus, by choosing n (and hence, n')

-1sufficiently small, it suffices to show that n T 1 + ~(F) a.s., as n~. Byn,

19

the same decomposition (i.e., (3.42», we can show that

Then, by (3.20) and (2.3), we have

(4.7)

Now, Xi = Qi¢(l) (G(Zi»' i>l, are iidrv with mean ~l(F), and hence, by the

Kintchine strong law of large numbers,

(4.8) a.s., as n-+oo.

Also, by theG1ivenko-Cante1li theorem, sup IG (z)-G(z) 1+0 a.s., as n-+oo, so thatz£.E n

on noticing that IQil ~ 1, V i.?::.l and ¢(1) is a polynomial, we immediately con-

clude that the second term on the rhs of (4.7), being bounded by

max 1¢(1)( ~1 G (Z.» - ¢(l)(G(Zi»I, converges a.s. to 0 as n-+oo. So, thel<i<n n n 1

.. proof is complete.

By (4.2), Lemma 4.1 and the definition of Bahadur (1960) efficiency [cf.

Puri and Sen (1971, p. 122)], we conclude that the BARE (Bahadur ARE) of {T }n

based on the score function ¢ with respect to {T*} based on the score functionn

¢* is given by

(4.9) e1 (¢,¢*) = [~(F,¢)A(¢*)/~(F,¢*)A(¢)]2,

00 1where ~(F,¢) = f n*(z)¢(G(z»dG(z), A2 (¢) = f ¢2(u)du and similar expression

-00 0for ~(F,¢*) and A2(¢*) hold for ¢=¢*. Notice that one may rewrite

(4.10)00

_00

00

I J n*(z)~(G(z»dG(z)]2

In* (z)] 2dG (z)}{ 1 _00 00 }

IJ ~2(u)du]I J In*(z)]2dG(z)]o -00

20

1 1 1 1= (J ~2(u)du){[J ~(u)~(u)du]2/[J ~2(u)du][J ~2(u)du]}

o 0 0 0

1= p2(~,~).(J ~2(u)du),

o

where ~(u) = n*(G-l(u» = 2n(G-l (u»-1), O<u<l. Thus, (4.9) reduces to

(4.11)

Thus, from the BARE point of view, the optimal choice of ~(u) is ~(u), O<u<l,

and as a result,

(4.12) e(~,~) = p2(~,~) is always bounded by 1.

We could have also considered the Pitman ARE, where we conceive of a

sequence {Hn} of alternative hypotheses, such that under Hn • F(x,y) = F(n) (x,y),

is such that Zl, .•• ,Zn are iidrv with a df G(n)(z) (dependent on n) and

n(z) = n(n)(z) also may depend on z, in such a way that

(4.13) ~:: G(n)(z) = G(z) exists, and n(n)(z) =~+n~y(z), z£E,

and J ly(z)1 1~(G(z»ldG(z)<oo. Then, if we let

(4.14)

(4.15)

1~*(u) = y(G-l(u», O<u<l, A(y) = J I~*(u)]2du;

o1

p(~*,~) = (J ~*(u)~(u)du)/[A(~)A(y)],. 0

it follows by some routine steps that the Pitman ARE of {T } with respect ton

{T*} isn

(4.16)

21

In this case, the asymptotically optimal score function is ¢=ljJ*.

consider the tests based + and M • that here also theLet us now on M Noten n

null hypothesis distribution + is generated by the 2n , equally likelyof M or M n.n n

realizations of (Q ,R). [It may be remarked that given Q and R , the vector-n -n -n -n

(Tl, .•• ,Tn) assumes a particular value dependent only on the score function and

(Q ,R ).] Thus, here, one can enumerate the distribution of M+ or M by direct-n -n n n

evaluation of all the 2nn! equally likely realizations of (gn'~n); by this

constitutio~, the statistics M and M+ are distribution-free under H. Then n .. 0

process of evaluating the exact null distribution of M+ or M becomes pro-n n

hibitively laborious as n increases. However, for large n, by virtue of Theorem

3.3 and well-known results on the boundary crossing probabilities for a standard

Brownian motion, we obtain that for every x>O,

(4.17)

(4.18)

lim P{M+ < xlH } = 2~(x)-1,n - 0-n-x>o

lim P{Mn ~ xlHo} = Ik:_oo(-1)k[~((2k+l)X)-~((2k-l)x)].n-x>o

Note that if W be the upper 100a% point of the df in (4.18), then by (2.8),a

(2.9), (4.17) and (4.18),

(4.19) lim M+ = T /2 and lim M = W :n-x>o n,a a n-x>o n,a a

O<a<l.

+If we denote the rhs of (4.17) and (4.18) by H (x) and H(x), respectively, we

note that by (4.17), for large x,

(4.20)

+ +Also, noting that l-H (x) ~ l-H(x) ~ 2[1-H (x)], we have for large x,

(4.21) -log[l-H(x)] = ~x2{1+o(1)}.

Further, by Lemma 4.1 and (3.12), it follows that as n-x>o

(4.22) ~+ ~n M + ~(F)/A a.s., and n 2M + I~(F)I/A a.s.n n

22

Hence, the efficacy [in the sense of Bahadur (1960)] of either M+ or M isn n

(4.23)1 1

~2(F)/A2 =([ ~(u)~(u)du)2/(f ~2(u)du),o 0

where ~(u) is defined after (4.10). As such the BARE of M+ (or M ) with respectn n

to T in (2.4 [or (2.5)] is equal to 1, when the same score function ~(u) isn

employed in both the cases. On the other hand, in (2.4)-(2.5), our sample size

is prefixed and equal to n, while in (2.8)-(2.9), it is a random variable N ,n

and N can be smaller than n with a positive probability. In fact, by Lemma 4.1n

and (4.19), it follows that for every £>0,

(4.24) P{Nn > £nl~(F)+O} ~ p{n~T[n£]/An > M:,al~(F)+O}

= p{n-lT[ ]/A > n~M+I~(F)+O}+O, as n+oo,n£ n n,a

and a similar result follows for M. Consequently, when H is not true, one mayn 0

expect a considerable amount of reduction of the ASN of the 1st sequential pro-

cedure, without any loss of the BARE.

RNvJ'WW.~;tJt~~k~)~M~&J~jW~>Jr, D: ~ Dn • Note that by Lemma 3.4, under

H , Q(S ) assumes all possible 2n realizations j £3 , each with the equalo - -n -n -n

probability 2-n • By a look at (2.12) and (2.14), we observe that the set of

realizations of (T 1, ••• ,T ), and hence, of D+ or D , generated by the set ofn nn n n

2n equally likely realizations of Q(S ), can be traced, and the exact null dis-- -n

tribution can be computed. By virtue of this constitution, the tests based on

D+ and D are distribution-free.n n

(5.1)

It follows from Theorem 3.6 that as n+oo,)

P{D+ < x} + p{sup Wet) ~ x},n - t£I

v O.s.x<oo;

(5.2)

As a result,

23

P{Dn ~ x} -+- p{sup IW(t) I ~ x}, V O<x<oo.tEl

D+ and M+ (or D and M ) both have the same limiting null distri-n n n n

bution given by (4.17) (or (4.18». Consequently, as in (4.19),

(5.3) lim D+ =T /2 and lim Dn+oo n,a a n+oo n,a

= W :a

O<a<l,

and (4.20)-(4.21) also apply to these statistics.

Let us now denote by

(5.4)X

T(X) = f TI*(z)¢(G(z»dG(z), -oo<x<oo.-00

Lemma 5.1. Under (3.17) and the conditions of Theorem 3.2,

(5.5) n~D+ -+- sup T(x) / A a. s. , as n+oo,n x

(5.6) n~D -+- sup IT(x) 1/A a. s. , as n+oo.n x

Proof. As in the proof of Lemma 4.1, we write, on using (3.42), Tnk =

Tnk,l + Tnk ,2 - Tnk ,3' Then, for j=2 or 3,

(5.7) I -1 I -1, k 2 ~ -1, k 2 ~max n T k' ~ (n [.'-lQ.) (n [.'-la . (i»l<k<n n ,J 1.- 1. 1.- n,J

~(k/n){ll.n1a2 .(i»~ < [lI. n1a2 j(i)]~n',- n 1.= n,J - n 1.= n,

where n'(>O) depends on n(>O) in (3.43), and n'-+-O as n-+-O. Consequently, it

suffices to replace Tnk by Tnk,l and ¢ by ¢(l) in D:, Dn and T(X), respectively,

where by (3.42), ¢(l) is a polynomial, and hence, (3.51) holds.

For some arbitrary £>0, choose a set of m (=m) points£

where £m > 1-£. We also, denote by£-

(5.9) knj = [nj£]+l, for j=l, ••. ,m, knm+l=n.

Note that TI*(z) is absolutely continuous and bounded, ¢(l) is a polynomial and

(5.10)

24

G is absolutely continuous with G(Zj)-G(Zj_l) ~ E, l~~l. Hence, for every

0>0, there exists an E>O, such that

IT(X)-T(y) I~o .for every x,ydZj_l,Zj], l~~m+l.

On the other hand, (3.17) insuring (3.69), and ITI*(z)l~l, imply that for every

k " l<k<n<k "nJ- - -.....:: nJ

(5.11) I -1 I -1\ q I "In (T l-T k 1) ~ n L'=k+l a 1(1)nq, n, 1 n,1 11

k1

~ {n- Li~k+l a~ 1 (i)}~ ~ {n- Li~~ an2 1 (i)}~

, nj-l 'jE 1

~ (f ~(l)(u)dU)~ < ~o, for every l~<m+l.(j-l)E

Consequently, it suffices to show that

(5.12) max I{ max [n-lT" lJ-T(Zj) 1+0, a.s., as n~.l<"<m+l l<i<k n1,~- -- nj

The proof of (5.12) follows along the lines of Lemma 4.1, and hence, the

details are omitted.

Let us now denote by

(5.13) T~ = sup T(X) and TO = sup IT(X)I.x x

Then, by (5.1), (5.2), (4.17), (4.18), (4.20), (4.21), Lemma 5.1 and (5.13),

it follows that the efficacy of D+ (or D ) in the sense of Bahadur (1960) isn n

given by

(5.14)

Here also, we note that if we let for -oo<x<oo,

(5.15)

= 0, otherwise,

and if we let

(5.16)

then we have

1= f 1jJ2(u)du,o x

-oo<x<oo

25

(5.17) (T~)2/A2 = sup [A~(1jJ)p2(1jJx'¢)]'X

where p2(1jJ ,¢) < 1. Note that A2 (1jJ) is non-decreasing, so that if we letx x

¢(u) = 1jJ (u) = ~*(G-l(u», O<u<l, the rhs of (5.17) is maximized; for any other00

¢(u) (not proportional to 1jJ (u», the rhs of (5.17) is bounded from above by" " 00

1" A~(1jJ) = I1jJ~(u)du, so that

o00

(5.18)

Hence, here also, maximizing the BARE leads us to the asymptotically optimal

sCore function 1jJ (u) = ~*(G-l(u», O<u<1. A similar result holds for D". In00 n

the next section, we shall study the optimal score function, in little more

details, for some important cases.

In some important special cases,

~*(z) can be written in more explicit forms, and the optimal score functions

can be obtained in simpler forms too.

6.1. Stochastically independent components. Here X and Yare stochastically

independent, so that for all (X,y)EE2

(6.1) F(x,y) = F(x,oo)F(oo,y) = Fl (x)F2(y), say.

Let f l and f 2 be the density functions for Fl and F2 respectively. Thenby

(3.1) and (3.2),

(6.2) g(z) = f l (z)[1-F2(z)]+f2 (z)[1-Fl (z)], ~(z) = f l (z)[1-F2 (z)}/g(z);

(6.3) ~*(z) = [fl (z) [1-F2 (z)]-f2 (z) (1-Fl (z)]]/(fl (z)(1-f2(z)]+f2 (z) [l-Fl(z)]]

26

where the hazard rates rl(z) and r 2 (z) are defined by

(6.4) ri(z) = f.(z)/[l-F.(z)], zEE, for i=1,2.1 1

Now, under Ho in (1.1), Fl =F2, so that rl(z) = r 2 (z) for all z. We consider

two special cases where Fl and F2 may differ in locations or scales. First

consider the model

Then r 2 (z) = r l (z-6), so that by (6.3),

For small 6, (6.6) yields (whenever rl(z) is differentiable)

(6.7) dr*(z) ~ (6/2)[dz log rl(z)], zEE.

Thus, for local translation alternatives, the asymptotically optimal score

function is

(6.8) ~oo(u) = [(d/dz) log rl(z)] -1 ,O<u<l.z=G (u)

We may recall that the classical two-sample location problem [viz., Hajek

and Sidak (1967, p. 66)], the locally most powerful rank test corresponds to the

score function

(6.9)

In general, (6.8) and (6.9) are different from each other. To show this, let

us consider the general exponential type of df's for which Fl , f l and fi exist

and the following hold:

(6.10) d~ { 1;:~~;) } + 0 as x+oo and d~ { :~~:~ J+ 0 as >+-00.

Note that (6.10) implies that -[l-Fl (x)]fi(x)!ff(x)+l as x+oo and Fl(x)fi(x)/ff(x)+l~

as X+OO, so that as x+oo,

(6.11)

and as x+-oo,

(6.12)

d~ log r i (x) = fi (x) /fl (x) + f l (x) / [l-F1 (x)]

= [fi(x)/fl(x)]{l+f~(x)/[l-Fl(x)]fi(x)}

= [fi (x)/fl(x)]{o(l)},

d { f 1 (x) f 1 (x) }dx log rl(x) = [fi(x)/fl(x)] 1 + l-Fl(x) • fi(x) •

{

flex) Fl(x) }= [fi(x)/fl(x)] .1+ l-F

l(x) • flex) [1+0(1)]

= [fi (x)/fl (x)] {l+Fl (x) [l-Fl (x) ]-l[l+o(l)]}

= [fi(x)/fl(x)]{l+o(l)}.

27

Thus, l/Joo(u) behaves alike 1jJ(u) as u+ 0, but differently when u+L In particular

for normal df, fi(x)/fl(x) = -x, so that it appears that l/Joo(u) attaches more

weight when u is small and less as u+l. From one point of view this is quite

important too. If the null hypothesis is not true, with greater weight for

small u, the Tnk will be crossing the barrierD+ (or + D ) faster than then,a - n,a

other case.where l/Joo(u) would have attached more weight to the upper tail. Thus,

we would ~xpect an early termination in such a case, and hence, the ASN for the

progressively censored test will be smaller when H does not hold.o

Consider now the scale model where

(6.13)

-1In this case, r 2 (z) = 8 r l (z/8), ZEE, so that

(6.14) .

and hence for 8=1+0, 0 small, (6.14) tends to

(6.15) (-0/2){1+z(d/dz) log rl(z)}, ZEE.

28

Consequently, for local scalar alternatives, the asymptotically optimal score

function is

(6.16) ~oo(u) = l+[z(d/dz) log r1 (z)] -1 ,0<u<1.. z=G (u)

By arguments similar to (6.10)-(6.12), it follows that (6.16) is generally

different from the optimal score function for the classical two-sample scale

problem.

6.2. Interchangeable components model. Here we assume that (1.1) holds under·

H and under alternative, X and Y-6 are interchangeable for some real 6. Thus,o

under alternative,

(6.17) F(x,y) = F (x,y-6), (x,y)£E2 ,o

where F (x,y) - F (y,x) for all (x,y). Let us denote the joint survival functiono 0

by

(6.18) F(x,y) = 1 - F(x,oo) - F(oo,y) + F(x,y), (x,y)£E2•

Then, note that under (6.17) and small 6,

00 00

(6.19) TI*(z) = 6[f (z,z) - J [(a/du)f (x,u)] dx]/[2J f (x,z)dx] + 0(6),o 0 u=z 0z z

where f is the density function corresponding to F. (6.19) reduces to (6.7)o 0

when f (z,z) = f2(z), V z£E. For specific f , such as the bivariate normalo 0 0

density, (6.19) may be evaluated and the corresponding ~(u) can be determined.

In general, these are quite complicated.

.29

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[4] DVORETZKY, A. (1972). Asymptotic normality for sums of dependent randomvariables. Proc. 6th Berkeley Symp. Math. Statist. Probe ~, 513-535.

~

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[6]~ V /

HAJEK, J. and SIDAK, Z. (1967).New York.

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HOEFFDING, W. (1973). On the centering of a simple linear rank statistic.Ann. Statist. 1, 54-66.

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LEE, L and THOMPSON, W.A., JR. Reliability of multiple component systems.Tech. Report No. 48, Mathematical Sciences, Univ. of Missouri,Columbia.

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