Statistics & Probability Letters 12 (1991) 233-237

North-Holland

September 1991

Testing the dispersive equivalence of two populations

Leszek Marzec and Paw& Marzec Mathematical Institute, University of Wroclaw, 50-384 Wroclaw, Poland

Received November 1990

Abstract: Based on independent random samples from the distribution functions F and G respectively, two invariant, unbiased and asymptotically distribution-free tests are proposed for testing the hypothesis Ha: F = d’spG against H,: F < d’spG. Asymptotic laws of

the test statistics under the null hypothesis are obtained.

Keywords: Partial ordering, spacings, two-sample problem, invariance, unbiasedness, isotonic power.

1. Introduction

Recently Lewis and Thompson (1981) Shaked (1982), Lynch, Mimmack and Proschan (1983) among others, have considered the following partial ordering on the space of probability distribution functions (d.f.‘s). For any pair of d.f.‘s F and G, F is said to be dispersed with respect to G, written as F < dispG, if

and only if

F-‘(P) -F-‘(a) < G-‘(P) - G-‘(a) whenever 0 <a < j3 < 1.

It is easy to see that F =dispG if and only if F(x) = G(x + p) for some p. Thus this partial ordering is location invariant. The purpose of this investigation is to provide a test statistic for testing H,: F =d’spG

versus H,: F < dispG and F # dispG, when two independent random samples are available from F and G respectively. The one-sample problem, i.e. when G is known, was considered by Bartoszewicz (1986) and Bartoszewicz and Bednarski (1990). The two-sample problem has not been discussed to the best of our knowledge.

The following notation will be used in the sequel. Given a random sample X = (Xi,. . . , X,), let xi:, < . . . < X,,:, denote the order statistics of X and DjTn = X,+l:n - X,:,, i = 1,. . . , n - 1, their corre- sponding spacings. If X and Y have d.f.‘s F and G respectively, then X <“Y means F(x) 2 G(X) for every x. After that it is assumed that each considered d.f. F is continuous and increasing on the interval whereO<F<l.

2. A property of the dispersive ordering

In this section we prove a useful property of the dispersive ordering which will be used in the sequel. Let X = (Xi,. . . , X,,) and Y = (Y,, . . . , Y,) be independent random samples from the d.f.‘s F, and G, respec-

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tively. Moreover, let the independent random samples W and Z be of size n and come from F, and G, respectively.

Theorem 1. If Fl < dispFZ and G, < dispG1, then

for every function 4 : R:-’ + R nondecreasing in each argument.

Proof. By assumption, F2- ‘F,(x) - x is nondecreasing in

X+,:n - x,:, G F,-%(x,+,x) - 5%(x,:,).

Similarly G, < d’sPGl implies

Y+,:n - Y:, 2 G,‘G,(K+,:,) - G,‘G,(K,).

x and consequently

Consequently for the function 4 nondecreasing in each argument we have

i

K’F,(X,:,) - F,-%(L) ” G,'G,(Y,:,) - G;'G,( Y,:,) ‘...’

K’F,bL.) -K’F,(X-I:,) G,‘GdY,:.) - G,‘GdL,:n)

But F;‘F,( X,:,) (G; ‘G,( y:,)) has the same probability distribution as the ith order statistic of the sample from F, (G,). Since X and Y are independent the proof is complete. 0

3. Testing the dispersive equivalence

Given two independent random samples X = ( X,, . . . , X,,) and Y = (Y,, let us consider the problem of testing the hypothesis

H,: F=dispG against H,: FG~‘“~G and FZdiSpG.

This problem remains invariant under the group 9 of transformations

x:=x,+a, Y,’ = .Y, + b 9 i=l,..., n, a, bElW.

Y,) from F and G respectively,

By the principle of invariance applied to the space of sufficient statistic any invariant test of H, versus H, is a function of the maximal invariant under 9, i.e. of the statistic (D,:,, . . . , Dnylyn, DE,, . . . , D,‘_,:,)_

Let 4 : IF!:- ’ + IF! be a function which is nondecreasing in each argument. Then in view of Theorem 1,

+(Q’?‘D,?,,~ . . . > D:,:,,/D,‘-,:.) tends to be smaller under the alternative hypothesis than under the null hypothesis, so that rejection of H, occurs for small values of this quantity. This suggests using a level cx test of the form

4(X> y> = 1 if +(D~,/D~.,..., Dn?l:./D,‘-l:.) <d,,,, (1) 0 otherwise,

where d, a ( is determined so that E,,+(X, Y) = (Y. We have:

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Theorem 2. Each test $I of the form (1) has isotonic power in the sense that F, < dispF2 <dispGZ < dispGl implies

%,.G$(X’ Y) G EF,.G,+(X, Y>.

Proof. It follows directly from Theorem 1. 0

Corollary. Each test $I of the form (1) is unbiased. 0

4. Tests based on maximum and minimum

Consider two tests (P, and +z of the form (1) which are defined by the functions

q,(z) = max z, and lc/Z (z ) = min z, l<i<n-1 l<i<n-1

respectively. In view of the results of Section 3 the tests +i and +Z are invariant, unbiased and have isotonic powers with respect to the dispersive ordering.

Let F be the class of all absolutely continuous d.f.‘s F with positive density f on the interval where 0 < F < 1, such that

8-a ,Xs;p<61# -ll=O, lim (2)

where rF(x) = fF-‘(x)/(1 - x), 0 < x < 1. Note that if rF is uniformly continuous and bounded away from zero then the above condition is clearly satisfied.

We shall prove the following theorem.

Theorem 3. Let FE 9. Then under the hypothesis Ha: F = dispG,

Proof. Let rF be defined in (2). Let T = F-‘K, where K denotes the exponential d.f., i.e. K(y) = 1 - eey, y 2 0. Obviously

D,fn = T(K-‘F(X,+I:,)) - T(K-‘F(XiY,,,)).

By using the mean value Lagrange theorem we obtain

W D,T,=(n-i+l)-‘;

rF(C?) ’ (3)

where IV, = (n - i + l)( K-‘F( X,+l:n) - K-‘F( Xju,,,)) and the random variable 0,, is such that F( Xii:,) < S,, < F(X,+,:,), i= l,..., n- 1. Analogously

D,~fi=(n-i+l)-‘A rA ’

where c = (n - i + l)(KP’G(Yj+,:,) - K-‘G(y:,)) and G(x:,) < 4, < G(x+,,,), easily seen that W and V are independent random samples from the exponential d.f. that under H,, rF= r,-, (3) and (4) imply

(4)

i=l ,...,n-1. It is K. In view of the fact

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where 2, = w:/I$ i = 1, _ . . , n - 1, are i.i.d. random variables such that P{ 2, <y } = y/(1 + y), y 2 0. Given a > 0, let

A,(a)= {(i+l)/n-F(X,:,)<u/n”*, F(Xj+l:n)-(i+l)/n<a/n’/2,

(i+l)/n-G(~~,)~a/n”2,G(~+,~,)-(i+l)/n~a/n”2, i=l,...,n-1).

(6)

Note that F(X,:.) and G(y:,) are distributed as the ith order statistics of the independent random samples from the uniform (0, 1) d.f. Let

%l =sup{r,(u): Iu-(i+l)/nI <a/n’/*},

lrrn=inf{rF(u): Iu-(i+l)/nl <a/n’/*}.

If we put

then in view of (5) and (6) we have

~~nnA.(a)cB,nA,(a)c~~nnA.(a).

Consequently, after some calculations we obtain

JP(B,)-e-““l~4(1-P(A,(a)))+P(~~)-P(~~)+jP(~~)-e-1’*I.

By applying Donsker theorem (see Billingsley, 1968) one may find for each E > 0 a positive number a = a(e) such that for all n, P( A,(a)) > 1 - E. Moreover

n-l

fQJ= l-I i l-

1

i=l xn (l/nx + r-,,,/e,) i ’

II-1

P(&)= n i

l- 1

I=1 1 xn(l/nx + L/,y,,) ’

Assumption (2) makes it obvious that for every fixed a > 0,

lim P( Fm) = lim P( En) = e-l/“. n-+m ?I+*

This completes the proof. q

Theorem 4. Let F E 9. Then under the hypothesis Ha: F = drspG,

The proof follows by the same method as in Theorem 3 and is omitted.

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From Theorem 3 and Theorem 4 it follows that for large n the tests $Q and & reject the hypothesis H, in favour of H, approximately at the level (Y if

-log(l - a) n

respectively. By Theorem 2 the tests with the above critical regions are asymptotically unbiased.

References

Bartoszewicz, J. (1986) Dispersive ordering and the total time

on test transformation, S~afist. Probab. Left. 4, 285-288.

Bartoszewicz, J. and T. Bednarski (1990) On a test for disper-

sive ordering, Srnrisr. Probub. Lerr. 10, 355-362.

Billingsley, P. (1968) Convergence of Probabiliry Measures (Wi-

ley, New York).

Lewis, T. and J.W. Thompson (1981). Dispersive distributions

and the connection between dispersivity and strong uni-

modality, J. Appl. Probab. 18, 76-90. Lynch, J., G. Mimmack and F. Proschan (1983). Dispersive

ordering results, Adu. Appl. Probab. 15, 889-891. Shaked, M. (1982) Dispersive ordering of distributions, J.

Appt. Probab. 19, 310-320.

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