Tests for a Multiple-Sample Problem in High Dimensions

Abstract

In this article, we consider the problem of testing the hypothesis on mean vectors in multiple-sample problem when the number of observations is smaller than the number of variables. First we propose an independence rule test (IRT) to deal with high-dimensional effects. The asymptotic distributions of IRT under the null hypothesis as well as under the alternative are established when both the dimension and the sample size go to infinity. Next, using the derived asymptotic power of IRT, we propose an adaptive independence rule test (AIRT) that is particularly designed for testing against sparse alternatives. Our AIRT is novel in that it can effectively pick out a few relevant features and reduce the effect of noise accumulation. Real data analysis and Monte Carlo simulations are used to illustrate our proposed methods.

Keywords:

• Asymptotic normality
• High dimension
• Hypothesis testing
• Multiple-sample problem
• Variable selection

Mathematics Subject Classification:

• Primary 62H15
• Secondary 62F05

Acknowledgments

We thank three anonymous reviewers for their helpful comments. This work was supported by the National Natural Science Foundation of China (Grant no. 11126260).