*J. Japan Statist. Soc.,* Vol. 37 (No. 1), pp. 53-86, 2007

M. S. Srivastava

**Abstract. **
In this article, we develop a multivariate theory for analyzing multivariate\break datasets that have fewer observations than dimensions. More specifically, we consider the problem of testing the hypothesis that the mean vector μ of a *p*-dimensional random vector *x* is a zero vector where *N*, the number of independent observations on *x*, is less than the dimension *p*. It is assumed that *x* is normally distributed with mean vector μ and unknown nonsingular covariance matrix Σ. We propose the test statistic *F*^{+} = *n*^{-2} (*p* - *n* + 1) *N*' *S*^{+}, where *n* = *N* - 1 < *p*, and *S* are the sample mean vector and the sample covariance matrix respectively, and *S*^{+} is the Moore-Penrose inverse of *S*It is shown that a suitably normalized version of the *F*^{+} statistic is asymptotically normally distributed under the hypothesis. The asymptotic non-null distribution in one sample case is given. The case when the covariance matrix Σ is singular of rank *r* but the sample size *N* is larger than *r* is also considered. The corresponding results for the case of two-samples and *k* samples, known as MANOVA, are given.

*Key words and phrases*:
Distribution of test statistics, DNA microarray data, fewer observations than dimension, multivariate analysis of variance, singular Wishart.

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