Hypothesis Testing in High-Dimensional Partially Linear Single Index Models

• Activity Number: 60 - Nonparametrics in High-Dimensional Data
• Type: Contributed
• Date/Time: Monday, August 3, 2020 : 10:00 AM to 2:00 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Shijie Cui* and Xu Guo and Runze Li
• Companies: and Beijing Normal University and Pennsylvania State University
• Keywords: High-dimensional; Hypothesis testing; Partial Penalization; Semiparametric regression; Sparsity

Abstract

This project is concerned with statistical inference for high dimensional partially linear single index models. Existing methods mainly focus on inference for high dimensional generalized linear regression model. The presence of high dimensional nuisance parameter and nuisance unknown function makes the inference problem very challenging. In this paper, we first propose a profile partial penalized least squares estimator and study its asymptotic properties. We propose a $F$-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 also introduce 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 tests. A real data example is used to illustrate the proposed testing procedures.