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

• Activity Number: 201 - Nonparametric Statistics Student Paper Competition Presentations
• Type: Topic-Contributed
• Date/Time: Tuesday, August 10, 2021 : 1:30 PM to 3:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Shijie Cui* and Xu Guo and Runze Li
• Companies: Pennsylvania State University and Beijing Normal University and Pennsylvania State University
• Keywords: High dimensional; Hypothesis testing; Semiparametric regression ; Penalized least squares; Local alternative; Sparsity

Abstract

This work is concerned with estimation and inference for ultrahigh dimensional partially linear single-index models. The presence of high dimensional nuisance parameters in semiparametric component and nuisance unknown function makes the estimation and inference problem very challenging. In this work, we first propose a profile partial penalized least squares estimator and establish the sparsity, consistency, and asymptotic expansion of the proposed estimator in an 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-squared distribution, with local alternatives converging to the null hypothesis at a root-n rate. We further propose a new test statistic with asymptotically normal distribution for the specification testing problem of the nonparametric function. Numerical studies are conducted to examine the finite sample performance of the proposed estimators and tests.