Multivariate Theory for Analyzing High Dimensional Data

In this article, we develop a multivariate theory for analyzing multivariate 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 ∑. We propose the test statistic F+ = n−2 (p − n + 1) N ¯x′S+¯x, where n = N − 1 < p, ¯x 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 ∑ 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.