Testing on the Multivariate Normal Covariance Matrix in High Dimensions

• Activity Number: 421
• Type: Contributed
• Date/Time: Tuesday, August 2, 2011 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistics in Epidemiology
• Author(s): Thomas Fisher*+
• Companies: University of Missouri at Kansas City
• Keywords: Hypothesis Testing ; Covariance matrix ; High-dimensional data analysis

Abstract

In this presentation we will discuss the problem of hypothesis testing on the covariance matrix when the dimensionality exceeds the sample size; a problem motivated by DNA microarray data. When in high-dimensions, the sample covariance matrix is singular and the likelihood ratio criterion is degenerate. Results in the literature recommend tests based on the arithmetic means of the eigenvalues of the covariance matrix as they are not perturbed by zeros. Much of the results rest on estimators for the first and second arithmetic means. We have developed unbiased and consistent estimators for the third and fourth arithmetic means of the eigenvalues. New test statistics are proposed under comparable assumptions to those statistics in the literature for the identity and sphericity hypotheses. The asymptotic distribution of the proposed test statistics are found in the general asymptotic framework. The statistics are shown to be consistent and comparable to those in the literature. A simulation study shows the newly proposed tests are effective and more powerful than those in the literature when just a few elements deviate from the null hypothesis.