SOME SCALE-INVARIANT TESTS FOR HIGH-DIMENSIONAL DATA

Abstract

Testing equality of mean vectors is a fundamental problem in high-dimensional data analysis. Traditional approaches such as Hotelling T-square or Lawley-Hotelling tests are no longer applicable. Several scale-invariant or non-scale-invariant tests have been proposed with normal distribution approximation, which requires strong assumptions on the covariance matrices. In this thesis, we propose and study several scale-invariant tests for testing equality of mean vectors for high-dimensional data, including two-sample problems under homoscedasticity and heteroscedasticity, and general linear hypothesis testing problems in high-dimensional regression. Our test statistics are simply constructed with their null distributions well approximated by the chi-squared-distribution without imposing strong assumptions on the covariance matrices. Ratio-consistent estimators of the parameters are obtained. Some theoretical properties of our test statistics, including approximate and asymptotic distributions and power functions, are established. Simulation studies and real data examples demonstrate the good performance of our tests, compared with several existing non-scale-invariant and scale-invariant tests.