U-Tests of General Linear Hypotheses for High-Dimensional Data Under Nonnormality and Heteroscedasticity

Abstract

Test statistics are presented for general linear hypotheses, with special focus on the two-sample profile analysis. The statistics are a modification to the classical Hotelling’s T2 statistic, are basically designed for the case when the dimension, p, may exceed the sample sizes, ni, and are valid under the violation of any assumption associated with T2, such as normality, homoscedasticity, or equal sample sizes. Under a few mild assumptions replacing the classical ones, the test statistics are shown to follow a normal limit under both the null and alternative hypothesis. As the test statistics are defined as a linear combination of U-statistics, the limits are correspondingly obtained using the asymptotic theory of degenerate (for null) and nondegenerate (for alternative) U-statistics. Simulation results, under a variety of parameter settings, are used to show the accuracy and robustness of the test statistics. Practical application of the tests is also illustrated using a few real data sets.