Estimation for a Partial-Linear Single-Index Model

TL;DR

This paper introduces a two-stage estimation method for a partial-linear single-index model, achieving efficient parameter estimation, optimal convergence rates, and facilitating inference, with validation through simulations and real data application.

Contribution

It proposes a novel two-stage estimation procedure that improves efficiency and convergence rates for the partial-linear single-index model, including extensions to multiple indices.

Findings

The estimator for the index is more efficient than existing methods.

The nonparametric link function estimator achieves optimal convergence rates.

Structural error variance can be accurately estimated.

Abstract

In this paper, we study the estimation for a partial-linear single-index model. A two-stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index, as well as the parameters in the linear component of the model. Asymptotic normality is established for both parametric components. For the index, a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance. The estimator of the nonparametric link function achieves optimal convergence rates; and the structural error variance is obtained. In addition, the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters. A simulation study is performed and an application to a real dataset is illustrated. The extension to multiple indices is briefly sketched.