Test for equality of standardized generalized variance with different dimensions under high-dimensional settings

Abstract

In this article, the testing equality of the standardized generalized variance (SGV) of $k$ multivariate normal distributions with possibly unequal dimensions is studied. The conventional likelihood-ratio test statistic reveals a serious bias with an increase in dimensions. Therefore, we present a new test statistic that eliminates this bias, and propose an asymptotic approximation-based test. Our proposed test is valid not only in high-dimensional settings but also in large sample settings. Additionally, we obtain the asymptotic non-null distribution of the proposed test and the approximate confidence interval of the SGV under high-dimensional and large sample settings. Finally, we investigate the finite sample and dimension behavior of this test using Monte Carlo simulations.