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

This article analyzes whether the existing tests for the p× p covariance matrix Σ of the N independent identically distributed observation vectors with N ≤ p work under non-normality. We focus on three hypotheses testing problems: (1) testing for sphericity, that is, the covariance matrix Σ is proportional to an identity matrix Ip; (2) the covariance matrix Σ is an identity matrix Ip; 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.