Tests for mean vectors in high-dimensional data

Abstract

High-dimensional data are ubiquitous and bring new challenges, not only to statisticians, but also to researchers in many scientific fields. They arise in situations where the dimension ( p) , the number of variables in a unit, is larger than the sample size (n), the number of units; data analysis using classical multivariate methods can no longer be applied.

In this study, the hypothesis testing problems considered are H : μ = 0 against K:μ≠0 in the one-sample case and H:μ1 =μ2 against K:μ1 ≠μ2 in the two- sample case; in both cases the data are of high dimension and assumed to be p- multivariate normal with unknown covariance matrix. In two-sample problems, the two samples are assumed to be independent and drawn from populations having a common covariance matrix Σ . The one-sample test statistic was developed based on the idea of keeping more, or as much information as possible, from the sample covariance matrix, after which the idea was extended to the two-sample case. The proposed test statistics, both for one- and two-samples, were shown to asymptotically follow a standard normal distribution when the dimension goes to infinity. One favorable property of the proposed tests is that they are invariant under a group of scalar transformations x→Dx, where D =diag(c ,...,c ) and c ≠0, for all i, 1pi i=1,...,p.Results from simulation studies compared the proposed tests with previously reported ones and showed that they performed acceptably well for all forms of covariance matrices under the study and achieved higher powers when the dimension increased for a given sample size. Applications of the proposed tests were illustrated using real-life DNA microarray data.