Some Tests Concerning the Covariance Matrix in High Dimensional Data

Abstract

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 (tr Σ i /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. Key words and phrases: Asymptotic distributions, multivariate normal, null and non-null distributions, sample size smaller than the dimension.