Neyman's C(α) test and Rao's efficient score test for composite hypotheses

Statistics & Probability North-Holland

Letters 11 (1991) 491-493

June 1991

Neyman’s C( a) test and Rao’s efficient score test for composite hypotheses S. Kocherlakota

and K. Kocherlakota

Department

University

of Statistics,

of Manitoba,

Winnipeg,

Man., Canada R3T 2N2

Received July 1990 Revised August 1990

Abstract: It is shown that Rao’s efficient used for the C(a) test.

score text and Neyman’s

1. Tbe problem Let the random variable X have the probability (density) function p(x; @), where 8 is a q-dimensional vector lying in the q-dimensional Euclidean space. Here the random variable is assumed to a scalar. This assumption is not restrictive, as a p-dimensional generalization follows quite easily. Let r(e)

= ?

log p(x,;

8)

(1)

i=l

and

i=1,2

,..-, 4, (2)

based on a random sample of size n. Also +

E

alog P(X;

i [ 0167-7152/91/$03.50

aei

e) a

log P(X;

ae,

69

Ii

0 1991 - Elsevier Science Publishers

(3)

C(a)

test are identical

when maximum

likelihood

estimates

are

is the information matrix based on a single observation. Rao (1973, p. 418) discusses the problem of testing the hypothesis H,:

R,(8)

=O,

i=l,2

,..., k,

(4)

with the usual restriction on Rj of admitting continuous derivatives of the first order. Among others, he suggests the efficient score test to test H,. This is based on the statistic S, = +*‘V*-p, where I$* ’ = {51(f3*),...,E,(t?*)} and V* is the information matrix determined for 8i,. . . , tl,, with 0 being replaced by 8 *, the restricted maximum likelihood estimators. Rao (1947) argues that the asymptotic distribution of the statistic S, is X2 on k degrees of freedom. Of late, much use has been made of the Neyman C(a) test. This test was developed by Neyman (1959) as a locally asymptotically most powerful (LAMP) test for composite hypotheses. Reference may be made to Subrahmaniam (1966) Tarone (1979) and Barnwal and Paul (1988) among others, for applications of this test in a wide range of problems. Akritas (1988) has given an asymptotic derivation of the test. In the next section we discuss this test in some detail.

B.V. (North-Holland)

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LETTERS

2. Neyman’s C(a) test

ferent forms. By definition test is based on the statistic

X, be a random sample of size n from Let Xi,..., p(x; e), 0 belonging to the parameter space 9. As before, we consider I( (3) = Zy=i log p(xi f?). What is of interest is the test of the composite hypothesis H,: 8i = S,” (specified), with 8i being the first k elements of 8. The remaining parameter set 0, (of q - k elements) is not specified. Let

S, = +*‘v*-&*,

Following Neyman, it can be shown that the vector +” has asymptotically, under Ho, the multivariate normal distribution with mean vector 0 and variance matrix vo=

a i0g l(e)

,*I2

w

6

,a..,

a i0g l(e) ae,

and =

E

alog P(x; 0) ae,

i [ x

alog P(X; e) ae,

Ill 8, = ep&

The statistic S, can be written as

alog aps(“;8) alog fp; 0)

E

i

score

where

y*

(5)

Rao’s efficient

[

I

J

Ill B,=@

(6) The procedure consists of finding the residual in the regression of (&, . . _, &) on ($i+ i, . . . , $). Thus, writing 17= # - B& where B = VAVi- , it can be shown that under Ho, 7 has the multivariate normal distribution in k dimensions, with mean 0 and the variance matrix I$ = Vi: vi;V$IP Hencef:nder Ho, the statistic

However, if we use the maximum likelihood estimator for the nuisance parameters in Neyman’s statistic N,, then the second term in (8) reduces to N,. Also, the first term is identically zero. Hence Rao’s efficient score statistic S, and Neyman’s C(a)-test statistic N, are actually identical under this estimation procedure for the nuisance parameters.

4. Discussion N, = PfGZI

(7)

is asymptotically distributed as a X2 with k degrees of freedom. Since in equation (7) the parameter vector 0, is unspecified, Neyman suggests the use of any fi-consistent estimators. In the next section it will be shown that Rao’s efficient score test and this test are identical.

The present paper draws attention to the fact that Rao’s efficient score test and Neyman’s C(a) test are identical when in the latter we use the maximum likelihood estimators for the nuisance parameters. This is important in view of the common usage of these estimators in practice. Also, it should be noted that Rao’s work predates that of Neyman by over a decade.

3. Relationship between the tests

Acknowledgements

Rao’s efficient score test and the C(a) test are essentially the same test expressed in slightly dif-

The authors wish to thank the Natural Sciences and Engineering Research Council of Canada for

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& PROBABILITY

References Akritas, M. (1988) An asymptotic derivation of Neyman’s C(a) test, Statist. Probab. Lett. 6, 363-361. Bamwal, R.K. and S.R. Paul (1988), Analysis of one-way layout of count data with negative binomial variation, Biometrika 75, 215-222. Neyman, J. (1959) Optimal asymptotic tests of composite

LETTERS

June 1991

statistical hypothesis, in: U. Grenander, ed., Probability and Statistics (Wiley, New York) pp. 213-234. Rao, CR. (1947), Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation, Proc. Cambridge Philos. Sot. 44, 50-57. Rao, C.R. (1973), Linear Statistical Inference and its Applications (Wiley, New York, 2nd ed.). Subrahmaniam, K. (1966), A test for ‘intrinsic correlation’ in the theory of accident proneness, J. Roy. Statist. Sot. Ser. B 28, 180-189. Tarone, R.E. (1979), Testing the goodness of fit of the binomial distribution, Biometrika 66, 585-590.

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