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 × p covariance matrix Σ 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 Σ is proportional to an identity matrix I p ; (2) the covariance matrix Σ 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 ) → ∞ , 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 ) .