A power contrast of tests for homogeneity of covariance matrices in a high-dimensional setting.

Multivariate statistical analyses, such as linear discriminant analysis, MANOVA, and profile analysis, have a covariance-matrix homogeneity assumption. Until recently, homogeneity testing of covariance matrices was limited to the well-posed problem, where the number of observations is much larger than the data dimension. Linear dimension reduction has many applications in classification and regression but has been used very little in hypothesis testing for equal covariance matrices. In this manuscript, we first contrast the powers of five current tests for homogeneity of covariance matrices under a high-dimensional setting for two population covariance matrices using Monte Carlo simulations. We then derive a linear dimension reduction method specifically constructed for testing homogeneity of high-dimensional covariance matrices. We also explore the effect of our proposed linear dimension reduction for two or more covariance matrices on the power of four tests for homogeneity of covariance matrices under a high-dimensional setting for two- and three-population covariance matrices. We determine that our proposed linear dimension reduction method, when applied to the original data before using an appropriate test, can yield a substantial increase in power.