A Random Projection Approach to Hypothesis Tests in High-Dimensional Single-Index Models

Abstract

In this article, we consider the problem of hypothesis testing in high-dimensional single-index models. First, we study the feasibility of applying the classical F-test to a single-index model when the dimension of covariate vector and sample size are of the same order, and derive its asymptotic null distribution and asymptotic local power function. For the ultrahigh-dimensional single-index model, we construct F-statistics based on lower-dimensional random projections of the data, and establish the asymptotic null distribution and the asymptotic local power function of the proposed test statistics for the hypothesis testing of global and partial parameters. The new proposed test possesses the advantages of having a simple structure as well as being easy to compute. We compare the proposed test with other high-dimensional tests and provide sufficient conditions under which the proposed tests are more efficient. We conduct simulation studies to evaluate the finite-sample performances of the proposed tests and demonstrate that it has higher power than some existing methods in the models we consider. The application of real high-dimensional gene expression data is also provided to illustrate the effectiveness of the method. Supplementary materials for this article are available online.

KEYWORDS:

• High-dimensional inference
• Hypothesis testing
• Random projection
• Single-index model

Supplementary Materials

The supplement contains the proofs of lemmas and theorems, theoretical results for partial tests, and additional numerical studies as well as R code.

Acknowledgments

The authors would like to thank the Editor, the Associate Editor, and the anonymous referees for their constructive comments that improved the quality of this article.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

This work is supported in part by the Research Grant Council of Hong Kong (15303319), and The Hong Kong Polytechnic University (P0030124, P0034285, P0042888).