Tests for high-dimensional single-index models

In this paper, we aim to test the overall significance of regres-sion coefficients in high-dimensional single-index models. We first reformu-late the hypothesis testing problem under elliptical distributions for pre-dictors. 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 re-spect to outliers in response. Moreover our procedure is free of variance es-timation of the error term. We establish the test statistic's asymptotic nor-mality 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 in-duced 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.