Empirical likelihood ratio test for or against a set of inequality constraints

journal of statistical planning and inference

Journal of Statistical Planning and Inference 55 (1996) 191-204

ELSEVIER

Empirical likelihood ratio test for or against a set of inequality constraints Hammou E1 Barmi

*

Department of Mathematics, University of Texas, Austin, TX 78712, USA Received 11 March 1994; revised 20 July 1995

Abstract We use the empirical likelihood ratio approach introduced by Owen (Biometrika 75 (1988), 237-249) to test for or against a set of inequality constraints when the parameters are defined by estimating functions. Our objective in this paper is to show that under fairly general conditions, the limiting distributions of the empirical likelihood ratio test statistics are of chi-bar square type (as in the parametric case) and give the expression of the weighting values. The results obtained here are similar to those in El Barmi and Dykstra (1995) where a full distributional model is assumed. This work presents also an extension of the results in Qin and Lawless (1995). A M S Classification: Primary 62E20 Keywords: Chi-bar square; Empirical likelihood; Estimating functions; Lagrange multipliers; Orthant probabilities

1. Introduction A problem which occurs frequently is that o f deciding on the basis o f a number o f independent observations o f a random variable whether a finite dimensional parameter belongs to a proper subset f21 o f the set ~ o f possible parameters. This problem has received considerable attention and a popular method for deciding the issue is the likelihood ratio approach. In this paper, we use instead the empirical likelihood ratio method introduced by Owen (1988). The justification for this is to consider a very large class o f possible distributions without having to specify a parametric model. The results obtained here are an extension o f those in Qin and Lawless (1995). This work is also closely connected with that o f E1 Barmi and Dykstra (1994a, 1995). Our objective is to show that empirical likelihoods have similar asymptotic distributions to those o f parametric likelihoods when testing for or against a set o f inequality constraints. To be * Tel.: (913) 532-6883; Fax: (913) 532 7736. 0378-3758/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0378-3758(95)00188-3

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H. El Barmi/Journal of Statistical Planning and Inference 55 (1996) 191-204

specific, suppose that X I , X 2 . . . . . X n are i.i.d, observations from an unknown distribution F and that there is a k-dimensional parameter 0 = (0l, 02..... 0k) x associated with F. We do not assume a full distributional model but we assume that information about F and 0 is only available through k-functionally independent functions ~p(x, 0) with E $(X, 0) = 0. Define ~2o and (21 as follows: Y2o : { 0 , 9 i ( 0 ) : 0, i : 1,2 ..... s}

and ~"~1

:

{O, gi(O)<.O,i = 1,2 . . . . . s},

where gi is a well behaved real valued function for all i and s<<.k. Based on a random sample of size n, we wish to use the empirical likelihood approach to test H0 versus HI - H0 and H1 versus H2 - H1 where Ho : 0 C (2o, H I : 0 C f21, and H2 imposes no restrictions on 0. We show under fairly general conditions that the asymptotic distributions of the empirical likelihood ratio test statistics corresponding to these cases are of chi-bar square type and give the expression of the weighting values. The rest of the article is organized as follows. In Section 2, we briefly describe the empirical likelihood as developed in connection with estimating functions in Qin and Lawless (1995), and in Section 3, we give the main results of this paper. Some examples and a simulation study are discussed in Section 4. Finally we give the proof of Theorem 3.1 in the Appendix.

2. Empirical likelihood and estimating functions The empirical likelihood method was introduced by Owen (1988, 1990) and was used to construct confidence intervals for vector valued functionals. It amounts to computing the likelihood of a multinomial distribution which has its atoms at data points. One of the remarkable contributions of Owen was to show a nonparametric version of Wilks' theorem. Empirical likelihoods have since been studied by many authors: DiCiccio and Romano (1989); DiCiccio et al. (1991); Hall and La Scala (1990) and Qin and Lawless (1995) being a small subset. In this section, we briefly describe the empirical likelihood in connection with estimating functions as developed in Qin and Lawless (1995) for testing H0 versus H2 - H 0 where H0 and H2 are as defined before. We do not make any assumptions about the form of the distribution function but consider parameters defined by estimating functions ~b(.,0), where E¢(X,O)

= O,

H. El Barrni / Journal of Statistical Planning and Injerence 55 (1996) 191 204

193

and q; is a k-dimensional vector. Since the parametric likelihood is not available, we will use instead the nonparametric likelihood

I-I Pi.

(2.1)

i--I

We: consider maximizing (2.1) with respect to pi, i - 1,2 . . . . . n and 0 subject to

£

Pi = 1,

pi>~O,Vi,

i--1

£

pi~9(Xi, O)

0

(2.2)

i--1

and the constraints in Hi, i = 0 or 2. For a fixed 0, (2.1) is first maximized with respect to pi, i the constraints

1,2 . . . . . n subject to

o. i

1

12.3)

i--I

If for a given 0, the convex hull of {¢(X1,0), ¢(X2, 0) . . . . . ~(X11, 0)} contains 0, then a simple Lagrange multiplier argument shows that the maximum occurs at the probability vector whose ith component is given by 1

pi = pi(O)= n(1 + tTO(Xi, O)) ' where t

i

1,2 . . . . . n,

(2.4)

t(O) is a k x 1 vector of Lagrange multipliers and satisfies

e(x,,0) i:, 1 + tf~p(X/,0) Plugging the values of

-

o.

(2.5)

pi's into (2.1) yields

Pi :

1 -t- tT¢(xi, 0)"

i=1

i=1

Ignoring the constant term, we define the empirical log-likelihood as tl

5~(0) = ~ ' log(1 + tT¢(xi, 0))

(2.7)

i=l

which is then minimized with respect to 0 subject to the constraints in Hi where i is either 0 or 2 to get what will be called the constrained maximum empirical likelihood estimator (MELE) o f 0 if i = 0 and the unrestricted MELE if i = 2. It is easy to see that 0 which minimizes (2.7) under H2 is the solution t o ~ ' l ~ / l ( X i , O) : 0 and that • ( 0 ) = 0 (Pi = 1/n in this case). Therefore the empirical log-likelihood ratio test statistic for testing H0 versus H 2 - H0 is given by T02 = 2min0~Hob¢(0) and converges (under appropriate regularity conditions) to a chi-square with s degrees of freedom under H0. For a more detailed discussion of this, see Qin and Lawless (1995). Next

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we shall extend the work of Qin and Lawless (1995) to testing for or against a set of inequality constraints. We show that the empirical likelihoods in this case have also similar asymptotic distributions to those of parametric likelihoods.

3. Main results In this section, we derive the limiting distributions of the empirical likelihood ratio test statistics for testing for or against a set of inequality constraints. It is seen that these limiting distributions are similar to those in E1 Barmi and Dykstra (1994a, 1995) where a full distributional model is assumed. Throughout the rest of the paper, we let 00 be the true value. We assume that it is possible to compute the estimates of 0 under different hypotheses introduced before when we do not assume a full distributional model, but consider parameters defined by estimating functions ~(., 0), where E q,(X, 0) = 0, and ff is a k-dimensional vector. In the discussion which follows, we need to introduce various assumptions concerning ff and g = (gl,g2 ..... gs) T. These assumptions are exactly those of Owen (1988) and Qin and Lawless (1995). 1. if(x, 0) is continuous and differentiable with respect to 0 in some neighborhood of 00. 2. EII~b(X, Oo)II3 < oo, E~(X, Oo)~b(X, Oo)T is positive definite and E{(0/00)~, (X, Oo)} is full rank. 3. The matrix (0/00)9(0) is of full rank and

02g(0) 0000 T

02 ~/t(X,0) 0000 T

exist and are bounded by some constant and some integrable function respectively in some neighborhood of 0o. Although we are primarily concerned with testing H0 versus Hi - H 0 and Hi versus H 2 - H1, we first consider testing H0 versus H~' - H0 where

H~

:

9i(00) = 0,

i = 1,2 . . . . .

S1,

with Sl ~
GI

Q1 R1 J '

(3.2)

H. El Barmi/Journal of Statistical Planning and Inference 55 (1996) 191-204

where

V =

195

E {~?#J(X,0o)/00} T { E ( 0 ( X ,0o)o(X,0o) T}-I E {c3O(X, 0o)/O0}, G

=

(Og(Oo)/30) and G1 is the sub-matrix of G corresponding to (91,92 ..... gs, ). Qin and Lawless (1995) proved that if 0o E Ho, then

v~(o*

-

o0)

=-PIE

-~-oO(X,Oo)

v~i=-QE

{ E ~ ( X , Oo)O(X, Oo)T}-ISn(.,Oo)+Op(1),

~-OqJ(X,Oo)

(3.3)

{EO(X, Oo)qJ(X, Oo)T} ISn(.,0o)+Op(1), (3.4)

and T

x/~ a~ = -Q1E I ~--~(X, Oo)} {E~(X, Oo)~I(X, Oo)T}-Isn(.,Oo)+Op(1), (3.5) where 0* is the empirical maximum likelihood estimator of 0 under H ~ , • and ~,* are the Lagrange multipliers corresponding to the constraints in Ho and HI respectively and

S,(.,Oo) = n -U2 ~ ~(xi, Oo).

(3.6)

i=1

We note that (3.3), (3.4) and (3.5) together with the previous conditions imply the following lemma. Lemma 1. If Oo E H0, then ~/-~(0~ -

0ol . . . . .

•

Ok -

00~,~, .....

~,,cq,

* ...

3~* )T

, s,

converges in distribution to a multivariate normal distribution with mean vector zero and covariance matrix -R

M

M T -R1

where M T = [-Ri,0]. Proof. Combining (3.3), (3.4) and (3.5) and using the Central Limit Theorem implies that ~/-n(O~

-- 001 . . . . .

• Ok - - O o k , ~ l . . . .

* . . . ~* )T ,~s,O~l, ' s,

converges under H0 in distribution to a multivariate normal distribution with mean vector zero and covariance matrix

P1 VP1 P1 VQ y P1 VQ~ ] QVPI QVQ T QVQTI I . Q1 VP1 QI vo T 01VQ T J

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H. El Barmi / Journal of Statistical Planning and Inference 55 (1996) 191-204

From (3.1) and (3.2) we can conclude that (i) P V + QTG = I, (ii) P G T = O, (iii) Q V + R G = O, (iv) QG T = I, (v) P 1 V + Q~G1 = I, (vi) P1G T = O, (vii) Q1V + R1 G1 : 0, (viii) Q I G T = I. It is easy to see that (v) and (vi) imply P1VP1 = P1, (iii) and (iv) imply QVQ T = - R and (vii), (viii) imply that Q1VQ~ = -R1. Also (vii) implies Q I V Q T = - R 1 G 1 Q T, but QG T = I, hence G1Q T = [I,0] so that Q1VQz = [ - R I , 0 ] = M. All this combined gives the desired conclusion. [] Let T0*1 denote the empirical log-likelihood ratio test statistic for testing H0 versus H T - H0, then

To*1 = 2

log(1 +tT~,(X/,0)) --

log(1 + t*T~b(Xi, O*))

,

i=1

where 0 in the MELE of 0 under H0 and ~ and t* are the values of t which correspond to 0 and 0* respectively. We have the following theorem whose proof is given in the Appendix. Theorem 3.1. I f Oo c Ho, then f o r any real number t we have 2 nlim ~ P( T~I >~ t ) = P(zs-s, >~t).

3.1. Testing H0 versus H1 - H o

We consider next testing H0 versus H1 - H 0 . Let T01 denote the empirical loglikelihood ratio test statistic corresponding to this situation. We show that under H0 the limiting distribution of T01 is a weighted mixture of chi-square distributions known as a chi-bar square distribution. The weights are expressed as sums of products of orthant probabilities of multivariate normals and are often called the level probabilities. Closed form expressions for these weights for small dimensions of the normal variables can be derived (Abrahamson, 1964). For higher dimensions, various approximation schemes have been developed for some special cases (Robertson et al., 1988). Let ~ denote the family of all subsets of {1,2 . . . . . s}. For n C ~ , let TOl(n) denote the empirical log-likelihood ratio test statistic for testing H0 versus H i ( n ) - H0, where HI(TZ): gi(Oo) : 0,

V i E 7z.

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197

We denote the MELE of 0 under Hl(~) by 0*(zr) and by ~*(~z) the vector of the Lagrange multipliers corresponding to the constraints in Hl(Tr). For 7r a proper subset of {1,2 . . . . . s}(~z # 0), define wi(O°) =

Z

P(N(O, ZI(Z~))~O)P(N(O,X(~z))>~O),

~, CardUz):j

where Z1 (re) = [G(zr) VG(zc)T]-1 and Z(~z) : G ( ~ c ) [ v - VGT(~)ZI(~z)G(~)V]GT(zrc), where G(Tz) is the submatrix of G with columns determined by the indices in zr and ~rc is the complement of ~z. We define wo(Oo) -- P(N(0, X0) >~0) and s 1

w,(00)

1 -

~-~wj(0o), j=0

where Xo GVG f. The wj's are thus sums of products of orthant probabilities (Kfido, 1963; Shapiro, 1988; Robertson et al., 1988). The key distributional result is given in the following theorem. Theorem

3.2. I f Oo ¢ H0, then f o r an3, real t, we have S

lim P(fol >~t) = Z

wj(Oo)P(z~_/>~ t),

n - - + c'x2,

j=O

wtiere Wj(00)'s are as defined above and )~g =- O. To prove Theorem 3.2, we make use of the following lemma (Shapiro, 1985). 3.3. Suppose X is N(0, I), P i~ an idempotent symmetric matrix q[" rank r and K is a cone defined by Lemma

K=

{ v E R k, d/Try<0,

i:

1,2 . . . . . k},

where di, i = 1,2 . . . . . k are vectors such that either Pdi = 0 or Pdi - di Jor all i. Then the conditional distribution o f X T P X , ,qiven X E K, is that o f a chi-squared random variable with r degrees o f J?eedom. A consequence of this lemma is that if X ~ N(O,Z) where Z is positive definite then p(xTx-1x>~t,

X>~O) : p ( x T z - - 1 X > ~ t ) P ( X > ~ O ) .

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198

Proof of Theorem 3.2. Let 0* be the MELE of 0 under H1. Then 0* equals 0*(zr) for exactly one ~ (Wollan and Dykstra, 1986). Moreover 0* = 0"(~) if and only if gi(O*(n)) < 0 for all i c rcc and c~*(rc)~>0 for all i C re. Therefore P(Tol ~>t)= ~

P(Tol >~t, O* = O*Or))

ncJ

P(Tol >~t, ~(~)~>0, Vi E ~z, gi(O*(Tr)) < 0, Vi Clrc).

= ~

Since Ooff Ho, 9i(0o) = 0 for all i. Using a Taylor expansion to second order, we can write k g/n(gi(0*(7~)) -- gi( O0) ) = g ~ ~-~ ~ j gi( O0)( O; ( 7~) -- Ooj ) ~- Op(l). j=l

Therefore

P(Tol>~t) ~P(Tol>~t, • ~

c~i 0z)~-0, Vi C 7z, gi(O*(TZ))--9i(O0)<~O, ViE~z c)

P(Tol(rC)>~t, x/nae*(Tr)/>0, v/nG(rcc)(0*(rc) - 00) q- o p ( l ) < 0 ) .

:rE~-

Using (3.3), (3.4), Lemma 1 and the same technique as in the proof of Theorem 3.1 we can conclude that

nlim P(Tol >~t)= Z p(UT(QTR-1Q - QT(Tz)R~l(rc)Ql(zt))u)t' 7rGff" -QI(/t)U>/0)

x P(-G(ztc)Pff~z)U<~O),

where U ~ N(0, V). Since (QTR-~Q - Q~(~)R~(rOQ~(~))VQT

= 0,

we can use Lemma 2 and Theorem 3.1 to conclude that lirn

P(Tol/>t) =

Z

P(Z~-c~rd(~)/> t)P(N(0, -R1 (tO) ~>0)

x P(N(0, G(~ c )el (rt) G(rcc))/> 0). A direct computation of Rl(r 0 and P1 (~) gives the desired conclusion.

[]

If it is the case that H1 is given by H1 "

gi(Oo)<~O,

i = 1,2 . . . . . r,

where r ~
lim P(T01 ~>t) = n--+o~

where

wj(Oo)'S are

~-~wj(Oo)P(Z2s_j)t), j=0

as defined before with ~ the family of subsets of {1,2 . . . . . r}.

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199

3.2. Testing Hi versus H2 - H 1 Next we consider testing H1 as a null hypothesis. We show that the results obtained by Wollan and Dykstra (1986) for the parametric case hold here. Let T~2 denote the empirical log-likelihood ratio test statistic for testing H1 versus H2 - H1. The key distributional result for this situation is given in the following theorem. Theorem 3.3. If Oo E H0, then for any real number t we have S

nlirno~P(T12 >~t) = Z Wj( Oo)P(z2 >7t), j=0

where the wy( Oo)'S are as defined before and Z2 =_ O. Proof. Let T12(Tr) denote the empirical log-likelihood ratio test statistic for testing HI(~) versus H2 -Hl(Tz). Qin and Lawless (1995) showed that under H0 T12(zr) = --n~I*T(Tc)RI-I(7z)~l*(7~)+ %( 1 ) so that

P(T12 >~t) = Z

P(T~2 >~t,O* = 0*(Tr))

rc C 3w

= ZP(T12(Tz)>~t, oe*(zr)>~O,gi(O*(zt)) < O,i E n c) rcE;~

= Z

P(-ni*T(zc)Rll(70°F(70 + op(l)~>t, x~o~*Oz) >~0,

v/nG0zC)(0*(:t) - 00) + %(1) < 0). Therefore, by Lemma 1, we have

P(UTQ~(Tt)R~(Tz)QI(rt)u >~t,Ql(z0U ~>0)

lira. P ( T n >~t) = ~ ~E:.~-

×P(G(~C)P1Uz)U ~<0), where U ~ N(O, V). Since

Q~(rOR-l(rC)Ql(Tz)VQ[(n) = Q[(Tr), we can use Lemma 2 to conclude that S

lira P(TI2 >~t) = ~_~ wj(Oo)P(z~ >~t), j=0 which is the desired conclusion.

[~

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H. El Barmil Journal of Statistical Plannin9 and Inference 55 (1996) 191~04

If actually 00 E H1, then one can argue that lira P(T12 i> t) = ~ wj(Oo)P(z~ ~ t), j-0

n .---+ o o

where wj(Oo)'s are as defined before with ~ and r being the family o f subsets o f {i,9(0o) = 0} and its cardinal respectively. Finally, suppose we wish to test H0 versus HT* - H0 where H~*"

gi(Oo) = O,

i = 1,2 . . . . . r,

9i(0o)~0,

i =- r ÷ l . . . . . s.

Let ~ * denote the family o f all subsets of {r + 1,r + 2 . . . . . s} and define the wj(O)'s as before with 7r E Y * and gc the complement o f g in {r + 1,r + 2 . . . . . s}. Following the same steps as in the proof o f Theorem 3.2, we see that if Tffl* is the empirical log-likelihood ratio test statistic for testing H0 versus H~'* - H0 and if 00 C H0, then for any real number t, we have s--r

lim P(T0*l* ~>t) = Z wj(O°)P(ZZ-r-J >~t), j-o where g 2 = 0. It is easy to show that s

r

lim P(T~I* ~>t) = lim Z WJ(O)P(ZZ--r--J >~t) n - - ~ o<3 n ---+ o ~ j-0 almost surely if 0 is a consistent estimator for 00 C H0. A natural estimator for 00 is the MELE under the restriction that 0 c H0. One can then use

s--r wj( O)p(z2 r_j > t ) j=O to compute the p-values. We note also that just as is the case when we have a parametric model, one can use the least favorable distribution (when it is known) to compute cutoff points for the tests under consideration.

4. Examples Example 1. Suppose X has a k-dimensional multivariate distribution with mean (0~,02 . . . . . Ok) and a known diagonal covariance matrix W whose diagonal is given by (l/w1, 1/w2 . . . . . 1/wk)(wi > 0, Vi). Take ~9(x,O) = x - 0 and suppose that we wish to test H0 versus H~ - H o where Ho and H1 are as defined before with gi(O) = Oi+l-Oi for i = 1,2 . . . . . k - 1, that is Ho :

0a = 02 . . . . .

Ok

H E1 BarmilJournal of Statistical Planning and lnlerence 55 (1996) 191 204

201

and

HI "

Oi ~ 0 2 <<. "'" ~. Ok.

It is easy to see that t/

&(.,Oo) = n l ' 2 ~ ( x ~ -

0o)

i=l

and V

W -1 .

Also, it is straightforward to see that w~-j(Oo) - P ( j + 1,k,w), j = O, 1. . . . . k - 1, are the level probabilities with weights w = (wl,w2 . . . . . wk) v (Robertson et al., 1988). Hence k

P ( j , k, w ) p ( z 2 _ l >~t),

lira.,~ P(7101 ~>t) = Z j 1

which is exactly the same result as in the parametric case. If actually W is unknown, then Theorem 3.6.1 of Robertson et al. (1988) can be used to obtain a conservative p-value. Example 2. Suppose (Xli,X2i,...,Xki) r, i = 1. . . . . n is a random sample of size n from a k-variate distribution with mean (01,02 . . . . . 0k) T and nonsingular covariance matrix. We wish to use the techniques developed in this paper to test H0 : 01 = 02 . . . . .

Ok

against H1

:

01~02~

-'-

~Ok,

It follows from E1 Barmi and Dykstra (1994b) that the empirical log-likelihood ratio test statistic for this case is given by

7~t=2

log i=1

\

l+~-'~tj(Xj+,,i-Xji)j=l

log ,=1

I+Zt;(X/~,j-X,i ,/=l

)

, //

where ([1, i2 . . . . . ik l) T solves

sup~log !,,w ~=~

I + Z tj(Xj+I, i - Xji) j=l

,

and (t~,t~ . . . . . t~_l) T solves the same optimization problem subject to tj>_>0, j 1,2 . . . . . k - 1 . E1 Barmi and Dykstra (1994b) have also developed an iterative algorithm to compute (tl, t2 . . . . . [k 1)T and (t]~, t~ . . . . . t;' 1)r.

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202

Table 1 Pretest and posttest data Subject Pretest 1

2 3 4 5 6 7 8 9 10

30 28 31 26 20 30 34 15 28 20

Posttest

Subject

Pretest

Posttest

29 30 32 30 16 25 31 18 33 25

11 12 13 14 15 16 17 18 19 20

30 29 31 29 34 20 26 25 31 29

32 28 34 32 32 27 28 29 32 32

Table 2 Simulation study n 20 40 60 80 100 p 0.059 0.066 0.052 0.049 0.052

Example 3. Table 1 from Moore and McCabe (1993) gives the pretest and posttest scores on the MLA listening test in Spanish for 20 high school teachers who attended an intensive course in Spanish. We hope to show that attending the institute improves listening skills. To do so, we use the techniques developed in this paper and the previous example to test Ho ' 01 = 02 against H1 • 01 ~ 02,

where 01 and 0 2 represent the means before and after attending the institute respectively. The value of the empirical log-likelihood ratio test statistic is 3.78 and the p-value based on the limiting distribution equal to p = (1/2)P(z~>>,3.78) ~- 0.025 while the value of the paired t-test is 2.025 and the exact p-value is 0.029. Example 4 (Simulation Study). Let T01 be the empirical log-likelihood ratio test corresponding to the situation discussed in Example 1 and assume k --- 2. We use a simulation study to investigate the convergence of T01 to a chi-bar square distribution under Ho. We simulate 2000 replications of size n each from a bivariate normal distribution with mean (0,0) T and covariance matrix I. We use n = 20,40,60,80,and 100. We use the technique discussed in the previous example to compute the values of the T01 and the proportion of times T01 exceeds 2.71, the 0.05 cutoff point of the chi-bar square distribution with one degree of freedom. The results are reported in Table 2 and are satisfactory for moderate sample sizes.

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203

Acknowledgements The author wishes to thank Professor Dykstra for helpful discussions on this and related problems and the referees for a detailed review and valuable comments.

Appendix P r o o f of Theorem 3.1. It follows from Qin and Lawless (1992) that T~I = - - n ~ T R - I ~ + n ~ * T R l l ~ * + O p ( 1 )

(E ~b~bT)-I Sn(.,Oo ), where E ~p~kT : E ~k(X, 0o) ~(X, 0o) T and E~/~30 : E ~ k ( X , 0o)/~0. Hence T~I converges in distribution to

uT(Q~R71Q1 - QTR-IQ)U, where U ~ N(0, V). Assume for now that

(Q~RllQ1 - QTR-1Q)V

(A.1)

is idempotent. We can conclude (Serfling, 1980) that Tffl converges in distribution to a chi-square distribution with rank (Q~R~ 1Q1 - QTR-I Q) degrees o f freedom. Since V is nonsingular and if the expression in (A. 1 ) is idempotent, it must be the case that rank (Q~R~IQ1 - QTR-1 Q) = trace (Q~R71Qa - QT R-l Q) V = trace (R~ 1Ql VQ~ ) - trace (R~IQVQ T) = S -- SI~

which gives the desired conclusion. Next we prove that the matrix (Q~Rl1Q1 - QTR-1Q)V is idempotent. It follows from Lemma 1 that R =- - Q V Q T and R1 = -Q1VQ~. Therefore

(Q~R~ 1QI - QTRQ) V(Q~R~Q1 - QTR-1 Q) = -QT1R~IQ1 - QR-1Q T _ QTR-1QVQTR~IQ1 - QTR~IQ~ VQTR-IQ.

204

H. El Barmi/Journal of Statistical Planning and Inference 55 (1996) 191-204

S i n c e b y (iii), R - ~ Q = - G V - 1 , a n d b y (i), Q T G = ( I - P V ) , w e h a v e QV R - l Q VQ~R ~ 1Q1 = - QT G Q ~ R - 1QI = -(I

- PV)Q~R-lQ~

= -O~R-IQ1, w h e r e the last i n e q u a l i t y h o l d s b e c a u s e P V Q ~ C o m b i n i n g all this g i v e s the d e s i r e d c o n c l u s i o n .

-- 0 (see the p r o o f o f L e m m a

1).

[]

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