Estimation and Inference in Ultrahigh Dimensional Partially Linear Single-Index Models

This paper is concerned with estimation and inference for ultrahigh dimensional partially linear single-index models. The presence of high dimensional nuisance parameter and nuisance unknown function makes the estimation and inference problem very challenging. In this paper, we first propose a profile partial penalized least squares estimator and establish the sparsity, consistency and asymptotic representation of the proposed estimator in ultrahigh dimensional setting. We then propose an $F$-type test statistic for parameters of primary interest and show that the limiting null distribution of the test statistic is $\chi^2$ distribution, and the test statistic can detect local alternatives, which converge to the null hypothesis at the root-$n$ rate. We further propose a new test for the specification testing problem of the nonparametric function. The test statistic is shown to be asymptotically normal. Simulation studies are conducted to examine the finite sample performance of the proposed estimators and tests. A real data example is used to illustrate the proposed procedures.