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

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## 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 *T*^{2} statistic, are basically designed for the case when the dimension, *p*, may exceed the sample sizes, *n*_{i}, and are valid under the violation of any assumption associated with *T*^{2}, 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.

## Keywords

Behrens-Fisher problem General linear hypothesis Profile analysis*U*-statistics

## AMS Subject Classification

62415## Preview

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