Likelihood Ratio Test For The Multivariate Normal Generalized Variance

Abstract

An interesting measure of variability in multivariate populations is the determinant of the covariance matrix Σp×p, denoted as |Σ|, commonly referred to as generalized variance. This measure succinctly captures the dispersion of a multivariate population into a single value, while accounting for inter-variable dependencies. Consequently, it finds applications across various domains concerned with assessing dispersion within multivariate populations of interest. In this study, we introduce a likelihood ratio test for the generalized variance of multivariate normal distributions, accompanied by a theoretical exposition on the distribution theory of sample generalized variances. We propose both the Likelihood Ratio Test (LRT) and the Bartlett-Corrected Likelihood Ratio Test (BCLRT) for assessing the hypothesis that the generalized variance equals a parameter η, where η ∈ R. The development of these tests is purely theoretical. Our recommendation is to employ the BCLRT test primarily in scenarios where p = 2, particularly when n > 30. As for the LRT test, we suggest its application in cases where p = 2 or p = 3, provided that n > 30, and for p = 5 when n > 50.