A dimension reduction based approach for estimation and variable selection in partially linear single-index models with high-dimensional covariates

In this paper, we formulate the partially linear single-index models as bi-index dimension reduction models for the purpose of identifying significant covariates in both the linear part and the single-index part through only one combined index in a dimension reduction approach. This is different from all existing dimension reduction methods in the literature, which in general identify two basis directions in the subspace spanned by the parameter vectors of interest, rather than the two parameter vectors themselves. This approach makes the identification and the subsequent estimation and variable selection easier than existing methods for multi-index models. When the number of parameters diverges with the sample size, we then adopt coordinate-independent sparse estimation procedure to select significant covariates and estimate the corresponding parameters. The resulting sparse dimension reduction estimators are shown to be consistent and asymptotically normal with the oracle property. Simulations are conducted to evaluate the performance of the proposed method, and a real data set is analysed for an illustration.