Tests for high-dimensional single-index models

Abstract

In this paper, we aim to test the overall significance of regression coefficients in high-dimensional single-index models. We first reformulate the hypothesis testing problem under elliptical distributions for predictors. Applying distribution-based transformation, we introduce a high-dimensional score-type test statistic. Notably, no moment condition for the error term is required. Our introduced procedures are thus robust with respect to outliers in response. Moreover our procedure is free of variance estimation of the error term. We establish the test statistic’s asymptotic normality under null hypothesis. Power analysis is also investigated. To further improve computational efficiency and enhance empirical powers, we also introduce a two-stage test procedure under ultrahigh-dimensional settings based on random data splitting. To eliminate the additional randomness induced by data splitting, we further develop a powerful ensemble algorithm based on multiple data splitting. We show that the ensemble algorithm can control the type I error rate at a given significance level. Extension to partial significance testing problem is also investigated. Lastly, numerical studies and real data analysis are conducted to compare with existing approaches and to illustrate the robustness and validity of our proposed test procedures.