*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 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.

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