ON TESTING A SET OF CORRELATION COEFFICIENTS FOR EQUALITY. I. SOME ASYMPTOTIC RESULTS.
Abstract
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 Statist. 34 149-151), and by Aitkin and Nelson (unpublished).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 Bartlett's and Lawley's 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)
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1967
- Accession Number
- AD0668151
Entities
People
- Leon Jay Gleser
Organizations
- Johns Hopkins University