Estimation and testing for partially linear single-index models

TL;DR

This paper develops efficient estimation and variable selection methods for partially linear single-index models, demonstrating their consistency, oracle properties, and effective model selection through theoretical analysis and simulations.

Contribution

It introduces semiparametrically efficient estimators, applies SCAD penalty for variable selection, and proposes consistent tuning parameter selection and hypothesis testing procedures.

Findings

SCAD estimators are consistent and have the oracle property.

BIC effectively identifies the true model.

Simulation studies confirm the methods' performance.

Abstract

In partially linear single-index models, we obtain the semiparametrically efficient profile least-squares estimators of regression coefficients. We also employ the smoothly clipped absolute deviation penalty (SCAD) approach to simultaneously select variables and estimate regression coefficients. We show that the resulting SCAD estimators are consistent and possess the oracle property. Subsequently, we demonstrate that a proposed tuning parameter selector, BIC, identifies the true model consistently. Finally, we develop a linear hypothesis test for the parametric coefficients and a goodness-of-fit test for the nonparametric component, respectively. Monte Carlo studies are also presented.