Some tests for the covariance matrix with fewer observations than the dimension under non-normality

This article analyzes whether some existing tests for the p x p covariance matrix Sigma of the N independent identically distributed observation vectors work under non-normality. We focus on three hypotheses testing problems: (1) testing for sphericity, that is, the covariance matrix Sigma is proportional to an identity matrix I(p); (2) the covariance matrix Sigma is an identity matrix I(p); and (3) the covariance matrix is a diagonal matrix. It is shown that the tests proposed by Srivastava (2005) for the above three problems are robust under the non-normality assumption made in this article irrespective of whether N <= p or N >= p, but (N, p) -> infinity, and N/p may go to zero or infinity. Results are asymptotic and it may be noted that they may not hold for finite (N, p). (C) 2011 Published by Elsevier Inc.