Asymptotic Distributions of Some Test Criteria for the Covariance Matrix in Elliptical Distributions under Local Alternatives

The asymptotic distributions under local alternatives of two test criteria for testing the hypothesis that the characteristic roots of the covariance matrix of an elliptical population, assumed distinct, are equal to a set of specified numbers, are derived. The two tests are the modified likelihood ratio test and a new test criterion proposed in this context for the normal model. Similar results are given for the two tests for testing that the covariance matrix is a specified positive definite matrix, in which case the two tests are the modified likelihood ratio test and a test proposed by Rao and Nagao for the normal model, and also for a test for the covariance structure in familial data, studied by Srivastava.