ON TESTING A SET OF CORRELATION COEFFICIENTS FOR EQUALITY. I. SOME ASYMPTOTIC RESULTS.
JOHNS HOPKINS UNIV BALTIMORE MD DEPT OF STATISTICS
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Consider a random p-dimensional vector x having a multivariate normal distribution. We are interested in testing the hypothesis H that the correlations rho sub ij between the elements of x are equal to a common value rho i not j. The likelihood ratio test of H versus general alternatives is difficult to evaluate and complicated in form. Alternative tests have been proposed by Bartlett J.R.S.S. Ser. B 16 296-298 by Lawley Ann. Math. Statist. 34 149-151, and by Aitkin and Nelson unpublished. The asymptotic null distributions of Bartletts and Lawleys tests have been obtained by Anderson Ann. Math. Statist. 34 122-148 and Lawley loc. cit.. The asymptotic null distribution of the Aitkin-Nelson test has not yet been obtained. The present paper obtains the asymptotic null distribution of the previously mentioned tests in a unified general fashion. Each of the above three tests is shown to be under H asymptotically equivalent to a member of a certain class of quadratic forms involving the sample correlations r sub ij. The asymptotic null distributions of such quadratic forms are obtained using the method of Lawley loc. cit.. The null distribution of the Aitkin-Nelson test is found to be dependent upon rho the parameter unspecified in the null hypothesis in such a fashion as to suggest that the Aitkin-Nelson test is unpractical for most applications. Author
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