Variable selection and direction estimation for single-index models via DC-TGDR method

This paper is concerned with selecting important covariates and estimating the index direction simultaneously for high dimensional single-index models. We develop an efficient Threshold Gradient Directed Regularization method via maximizing Distance Covariance (DC-TGDR) between the single index and response variable. Due to the appealing property of distance covariance which can measure nonlinear dependence between random variables, the proposed method avoids estimating the unknown link function of the single index and dramatically reduces computational complexity compared to other methods that use smoothing techniques. It keeps the model-free advantage from the view of sufficient dimension reduction and requires neither predictors nor response variable to be continuous. In addition, the DC-TGDR method encourages a grouping effect. That is, it is capable of choosing highly correlated covariates in or out of the model together. We examine finite-sample performance of the proposed method by Monte Carlo simulations. In a real data analysis, we identify important copy number alterations (CNAs) for gene expression.