Some Tests Concerning the Covariance Matrix in High Dimensional Data

In this paper, tests are developed for testing certain hypotheses on the covariance matrix Σ, when the sample size N = n + 1 is smaller than the dimension pof the data. Under the condition that (tr Σi⁄p) exists and > 0, as p → ∞, i =1,…,8, tests are developed for testing the hypotheses that the covariance matrix in a normally distributed 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.