Linear Hypothesis Testing in Linear Models With High-Dimensional Responses

Abstract

In this article, we propose a new projection test for linear hypotheses on regression coefficient matrices in linear models with high-dimensional responses. We systematically study the theoretical properties of the proposed test. We first derive the optimal projection matrix for any given projection dimension to achieve the best power and provide an upper bound for the optimal dimension of projection matrix. We further provide insights into how to construct the optimal projection matrix. One- and two-sample mean problems can be formulated as special cases of linear hypotheses studied in this article. We both theoretically and empirically demonstrate that the proposed test can outperform the existing ones for one- and two-sample mean problems. We conduct Monte Carlo simulation to examine the finite sample performance and illustrate the proposed test by a real data example.

Keywords:

• Hotelling T² test
• Multiple sample mean test
• Projection test
• Two-sample mean test

Supplementary Materials

The online appendix consists of proofs of theorems in the paper. The supplementary materials consist of additional simulation results and technical details for proofs in the online appendix.

Acknowledgments

The authors are indebted to the referees, the associate editor, and the co-editor for their valuable comments, which have significantly improved the article. The content is solely the responsibility of the authors and does not necessarily represent the official views of NSF and NIH.

Additional information

Funding

This research was supported by NSF grants DMS 1820702, DMS 1953196, DMS 2015539 and NIH grants NIAID 5-R01-AI136664, R01CA229542, and R01 ES019672.