Statistics & Probability Letters 11 (1991) 149-153

North-Holland

February 1991

Locally most powerful test for testing the equality of variances of two linear models with common regression parameters

Manzoor Ahmad University of @_&bee, MontrCal, Quebec, Canada H3C 3P8

Yogendra P. Chaubey Department of h-fathematm and Statistics, Concordia University, 7141 Sherbrooke St West, Mont&al, Quebec, Canada H4B lR6

Received April 1989

Revised February 1990

Abstract: In this paper, the problem of testing the equality of two homoscedastic normal linear models with common regression

parameters is considered. A locally most powerful test which is invariant with respect to the group of location and scale

transformations of the observations is derived. The test statistic when simplified reduces to the ASR test statistic proposed and

studied by Chaubey (1981). The robustness of this test is further explored.

Keywords: Heteroscedasticity, LMP test, elliptically symmetric distributions. ASR test

1. Introduction

Consider two linear models given by

x=X,/3+&,, i=l,2, (1.1)

where r; is an n, x 1 observation vector, X, is an n, X k known non-stochastic matrix with r( X,) = k

(n, 2 k), ,8 is a k X 1 regression parameter, and E,, the n, X 1 disturbance vector, is normally distributed with zero mean and dispersion matrix g(~,) = u,~Z~,, and Ed and ~~ are independently distributed. Such models have been extensively studied under the paradigm of Bayesian analysis (see Box and Tiao (1973, Chap. 9)). We are interested in testing H, : uf = u 2’. This framework was considered by Chaubey (1981) who proposed and stadied a test based on the ratio of sums of squares of ordinary least squares residuals. This test was called the ASR test. The test was shown to have a monotone power function for two special cases; (i) the case when X, = X2, and (ii) the case when X, is simply a column of ones, a case considered by Geisser (1965). The power functions were numerically compared for the latter case with some alternative tests and a suggestion was made that the ASR test seemed to be locally powerful.

No optimal tests for this problem have been derived in the literature. The problem may be considered arising out of testing the heteroscedasticity of a linear model where the regression model is decomposed

This work was partially supported by NSERC Grants Nos. A3661 and A3450

0167-7152/91/$03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland) 149

Volume 11, Number 2 STATISTICS & PROBABILITY LETTERS February 1991

into two linear models as in (1.1) according to some criterion. Harrison and McCabe (1979) have proposed a test which is similar to the ASR test using this approach for testing the heteroscedasticity of a linear model. The problem in this form has attracted a lot of research but the optimal solution is yet to be obtained.

In this paper, however, we demonstrate the local optimality of the ASR test for the problem stated above. It may be remarked here that the UMP test does not exist here because of the non-existence of a complete sufficient statistic. In Section 2, the locally most powerful invariant test is derived and it is observed to be the ASR test. Section 3 explores its robustness properties.

2. Locally optimal test for H,

We write model (1.1) in a compact form as

Y=Xp+&,

where

Y’ = (Y,‘, r;>, X’= (X,‘, X2/), E’ = (E;, E;),

with

(2.1)

Let e = QY, where Q = Z - X( X’X))‘X’, denote the ordinary least square residuals. The ordinary least square residual vectors for the two models are given as e, = Q,,F + QlzY2 and e2 = Q2,G + Q2*yZ, where Q,, (i, j = 1, 2) is obtained by partitioning Q such that Q,, is an n, X n, matrix. Specifically, Q;, = Z - P,,,

where P,, = X,(X/X)-IX,‘. Now consider the group G = (SC.,) where g,.,(Y) = c(Y - Xy), with c a

positive constant and y a k x 1 vector. The problem of testing H, is invariant under G. Now we are ready to state the following theorem.

Theorem 2.1. The locally most powerful invariant test (under the group of location and scale changes) for Ha aguinst H, : 6’ > 1 depends on the statistic

T = eie,/( e;el + e;e2), (2.2)

which rejects H, in f&our of H, if it is small.

Proof. Since the distribution of any invariant test statistic depends only 8, the parameter maximal invariant, we may assume without loss of generality that p = 0, uf = 8, and cr,’ = 1. A left invariant

measure on the group G = lF4 +X [w k is ck-i dy dc. Using Wijsman’s (1967) representation theorem, the

ratio R of the non-null distribution to the null distribution of a maximal invariant is given by

zc 0-42

/I n+kpl exp{ -c’( Y - Xy)‘D( 19)( Y - Xy)/20} dy dc

R= 0 2

(2.3) 10

// cntk-’ exp{ -c’(Y-Xy)‘(Y-Xy)/2} dydc

0 Rk

where

Z D(e) = [ I (;’ eF

n2

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(2.4)

Note that for any vector b by usual least squares results we have

(Y-Xh)‘D(B)(Y-Xb)=U(B)-t(?-h)‘X’D(B)X(?-h),

where

p= (X’D(B)X)_‘X’D(B)Y and V(O) = (Y- Xq)‘D(B)(Y- XT).

Hence, the exponential term in the numerator of R becomes (- c*U( 6) - c*( 7 - y)‘X’D( 6)X( 7 - y))/28. Now, using the standard multivariate normal and gamma integrals we can write the numerator of (2.3) as

r( n/2)2 (n/2)-1(21T)~/2~(n2+k)/*1X~~(e)XI-”2(~(e))-n/2,

The denominator of (2.3) can similarily be written as

I?n/2)2 (“/2)w(271)w2) , ,y’x,-~/y&~“/*

and therefore any invariant test depends only on the ratio T, = U(O)/(e’e). It is clear from the above exposition that no UMPI test for this problem exists. To get the LB1 test we expand r, in the powers of (0 - 1) and consider the linear term as the test statistic. Towards this goal we write p as

~=(x~x+(e-1)x;x,)~‘(x’~+(e-1)~;~,)

and use the fact that (A + qB)-’ =A-’ - qA-‘BA-’ + o(q) to get

(P-P)=(e-1)(xX-‘~;~,+~(e-i),

where p^ = (X’X))‘X’Y. Using (2.4) again we have

e’e+(e-l)e;e,=(Y-xp^)‘o(e)(Y-xp^)=U(e)+(P^-~)’x~o(e)x(P^-P),

and hence using (2.5) we get

(2.5)

u(B)=e’e+(8-1) e;e2 - (e - 1)2e;X2( X’X)p’X’D(r3)X( X’.X)-‘X&, + .((e - I)‘)

=e’e+(&l)e;e,-(e-1) 2e~X2(X’X)-‘X,‘e2+o((e-1)2).

Therefore,

T,=l-t(8-1) e;e,/e’e+o(O-l)=l+(e-l)T+0(e-1).

Since the ratio of densities is a monotonic function of T,-‘, the LB1 test for H, vs. H, rejects H, for small values of T and the result follows. 0

Corollary 2.2. The locally optimal test derived above is equivalent to the ASR test.

Proof. The ASR test statistic is given by T, = e;e,/e;e,. Since T, = (1 + T,)-’ which is monotone in T2,

the statement is obvious. q

3. Robustness of the ASR test

Now we shall briefly discuss the robustness of the above test when the probability density function (pdf) of Y = (Y,‘, Y,‘)’ is given by an elliptically symmetric density

~(KP, e)=e-~~‘24(o;-X~)‘~(e)(Y-Xp)/e}, (3.1)

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Volume 11. Number 2 STATISTICS & PROBABILITY LETTERS February 1991

where q( .) satisfies the condition

s q(u’u) du=l.

R ” (3.2)

Note that q( .) need not be exponential so that the joint distribution of Y, and Y, is more general than normal. For testing H, : 0 = 1 vs. H, : 6 > 1, note that this problem is invariant under group G as has been pointed out above. Proceeding in the same manner as in the previous section, the ratio of non-null to null distributions of a maximal invariant under group G (taking p = 0 again without loss of generality) is given

by ra

lJ / “1’2cn+k-‘q{c2(y-Xy)‘D(B)(y-X-y)/B} dy dc

0

cc

cnfkp’q{c2(y-Xy)‘(y-Xy)} dydc (3.3)

JJ 0 R”

Simplifying the argument of the function q( .) by using (2.4) as in the previous section we can write the numerator of the above ratio as

@/‘I X’D( 6) XI ~“2~mP’1/2c”-1q* (c*U( 6’)/6’) dc,

where q*(x) = lw”q( u’u + x) du. The above expression further simplifies to

@n,+k)‘21 x’o(e)xl ~“2(u(e))-“~2Jo’q*(w~)w”~~ du

Analogously, the denominator of the ratio in (3.3) is given by

IX’X/~“2(e’e)~“/2~aq*(UZ)U”~’ dw.

The above shows that the LB1 test is the same as the test derived under the normal model. Moreover, this test statistic- 7J Y), say-satisfies the following properties:

(1) T((Y- XY)E~“~) = T(Y) V y E R and VE p.d.,

(2) T(aY) = T(Y) v’a > 0.

Hence, the test derived here is null robust (see Kariya (1981)). Also, since the above test does not depend on q(e), any null robust invariant test is non-null robust. Furthermore, the optimum test based on the statistic derived above is optimality robust since it is independent of q( .). The null robustness of this test makes it a proper candidate for testing heteroscedasticity as it was used in Harrison and McCabe

(1979).

Acknowledgement

The authors are grateful to B.K. Sinha for his comments. The comments of an anonymous referee also helped to improve the presentation of the paper.

References

Box, G.E.P. and G.C. Tim (1973), Bayemm Inference m Statm

trccrl Analysrs (Addison-Wesley, Reading, MA). Chaubey, Y.P. (1981), Testing the equality of variances of two

linear models, Canad. J. Statist. 9, 119-127.

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Volume 11. Number 2 STATISTICS & PROBABILITY LETTERS February 1991

Geisser, S. (1965), A Bayes approach for combining correlated estimates, J. Amer. Statist. Assoc. 60, 602.

Harrison, M.J. and B.P.M. McCabe (1979). A test of hetero-

scedasticity on ordinary least square residuals, J. Amer.

Statist. Assoc. 74, 494-499.

Kariya, T. (1981), Robustness of multivariate tests, Ann. Statist.

9, 1267-1275.

Wijsman, R.A. (1967). Cross-sections of orbits and their appli- cation to densities of maximal invariants, in: Proc. 5th

Berkeley Symp. on Mathematwal St&stirs and Probabrlq,

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400.

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