Threshold single index regression model from high-dimensional data

Abstract

Instability of model parameters is a common phenomenon in areas such as signal processing, quality control, fault detection, finance, security, and clinical medicine. In this paper we focus on the high-dimensional single index model with a possible change point due to a covariate threshold. We propose the $\ell_1$-penalized likelihood estimators for regression coefficients as well as the threshold parameter. A proximal gradient algorithm is proposed for detecting the possible change point. Besides, non-asymptotic Oracle inequalities for both the prediction risk and the $\ell_1$-estimation loss for regression coefficients are obtained under certain sparsity condition.At last, thorough simulation studies are conducted to illustrate the empirical performance of the proposed method.