J. Japan Statist. Soc., Vol. 35 (No. 2), pp. 251-272, 2005

Some Tests concerning the Covariance Matrix in High Dimensional Data

Muni S. Srivastava

Abstract. 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 p of 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.

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

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