LINEAR HYPOTHESIS TESTING FOR HIGH-DIMENSIONAL DATA UNDER HETEROSCEDASTICITY

Abstract

The Behrens-Fisher problem is one of the most fundamental problems in Statistics. In this thesis, we mainly consider three high-dimensional hypothesis testing problems: the two-sample Behrens-Fisher problem, the heteroscedastic one-way MANOVA, and the general linear hypothesis under heteroscedasticity. Although these problems have been thoroughly studied in the classical setting, high-dimensional data make most classical methods invalid due to the singularity of the sample covariance matrix, and new tools and techniques are in demand. We propose L2-norm type tests with Welch-Satterthwaite chi-squared approximation and a U-statistic based test using normal approximation to solve these high-dimensional Behrens-Fisher problems. Our tests are simple in form and are computationally efficient, and hence are easy to implement and suitable for large scale real data analysis. Theoretical properties of our tests, such as asymptotic normality and power, are established. Simulation studies and real data examples are also presented to demonstrate the good performance of our tests.