Test for equality of generalized variance in high-dimensional and large sample settings

Abstract

Generalized variance (GV), proposed by Wilks [8], is a one-dimensional measure of multidimensional scatter. It plays an important role in both theoretical and applied research on analyzing big data. This article examines the problem of testing equality of generalized variances of $k$ multivariate normal populations in high-dimensional and large sample settings. The conventional likelihood-ratio test statistic reveals a serious bias as dimensions increase. We present a new test statistic that eliminates this bias, and propose an asymptotic approximation-based test. The likelihood-ratio test statistic can be interpreted as an estimator of criteria related to Jensen’s inequality. Our test statistic is obtained by appropriately estimating this criteria in high-dimensional and large sample settings. In addition, our proposed test is valid not only in high dimensional settings but also in large sample settings. We also obtain the asymptotic non-null distribution of the proposed test in high-dimensional and large sample settings. Finally, we study the finite sample and dimension behavior of this test through Monte Carlo simulations.