Some Tests Concerning the Covariance Matrix in High Dimensional Data

Abstract

Recent advances in analyzing high dimensional data with fewer observations require that certain assumptions made implicitly or explicitly in analyzing them should be ascertained. For example, Dudoit et al. (2002) in their analysis of microarrays data on genes assume that the covariance matrices are diagonal matrices, and thus their distance function uses only the diagonal elements of the sample covariance matrix. The good performance of their procedures appear to suggest that this indeed might be the case. To ascertain these assumptions on the covariance matrix, the likelihood ratio tests cannot be used as the sample size N = n+1 could be smaller than the dimension p. Although, the locally best invariant (LBI) test proposed by John (1971) and considered by Suguira (1972), and Nagao (1973) for the spherecity hypothesis and the LBI test given by Nagao (1973) for testing the hypothesis that the covariance matrix Σ is an identity matrix can be computed for all sample sizes, there appears to be no theoretical justification for using them as LBI tests cannot be obtained when n<p . Thus, we consider a distance function between the null hypothesis and the alternative hypothesis, and propose tests based on consistent estimators of these parametric functions of the covariance matrix Σ for testing the hypothesis of spherecity of the covarinace matrix, and for testing the hypothesis that the covarinace matrix is an identity matrix. In addition, under the same set of conditions, we provide tests for testing the hypothesis that the covarinace matrix is a diagonal matrix. Asymptotic distributions of these test statistics are given under the hypothesis as well as under the alternative hypothesis. Our focus is however for the case when n = O(p δ ), 0 <δ ≤ 1. Thus, it includes the case when (n/p) → 0. The