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×pp×p covariance matrix ΣΣ of the NN 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 ΣΣ is proportional to an identity matrix IpIp; (2) the covariance matrix ΣΣ is an identity matrix IpIp; 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≤pN≤p or N≥pN≥p, but (N,p)→∞(N,p)→∞, and N/pN/p may go to zero or infinity. Results are asymptotic and it may be noted that they may not hold for finite (N,p)(N,p).