Some tests concerning the covariance matrix in High-Dimensional data

In this paper, tests are developed for testing certain hypotheses on the covari-ance matrix Σ, when the sample size N = n+1 is smaller than the dimension p of the data. Under the condition that (trΣi/p) exists and> 0, as p→∞, i = 1,..., 8, tests are developed for testing the hypotheses that the covariance matrix in a normally dis-tributed data is an identity matrix, a constant time the identity matrix (spherecity), and is a diagonal matrix. The asymptotic null and non-null distributions of these test statistics are given. Key words and phrases: Asymptotic distributions, multivariate normal, null and non-null distributions, sample size smaller than the dimension. 1