Statistical Inference for High-Dimensional Heteroscedastic Partially Single-Index Models.

In this study, we propose a novel penalized empirical likelihood approach that simultaneously performs parameter estimation and variable selection in heteroscedastic partially linear single-index models with a diverging number of parameters. It is rigorously proved that the proposed method possesses the oracle property: (i) with probability tending to 1, the zero components are consistently estimated as zero; (ii) the estimators for nonzero coefficients achieve asymptotic efficiency. Furthermore, the penalized empirical log-likelihood ratio statistic is shown to asymptotically follow a standard chi-squared distribution under the null hypothesis. This methodology can be naturally applied to pure partially linear models and single-index models in high-dimensional settings. Simulation studies and real-world data analysis are conducted to examine the properties of the presented approach.