MULTIVARIATE THEORY FOR ANALYZING HIGH DIMENSIONAL DATA

概要

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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 Σ. We propose the test statistic F^+=n^<-2>(p-n+1)Nx^^-'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 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.