Semiparametric GEE analysis in partially linear single-index models for longitudinal data

TL;DR

This paper develops a semiparametric GEE approach for partially linear single-index models in longitudinal data, addressing both sparse and dense data scenarios with theoretical and empirical validation.

Contribution

It introduces a novel estimation method combining local linear smoothing with GEE for longitudinal data analysis in partially linear single-index models.

Findings

Estimates have different convergence rates for sparse versus dense data.

Asymptotic properties of estimators are derived under mild conditions.

Numerical studies confirm the effectiveness of the proposed method.

Abstract

In this article, we study a partially linear single-index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases. A semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations (GEE) is introduced to estimate the two parameter vectors as well as the unknown link function. Under some mild conditions, we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios, from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data. We also discuss the estimation of the covariance (or weight) matrices involved in the semiparametric GEE method. Furthermore, we provide some numerical studies including Monte Carlo simulation and an empirical application to illustrate our methodology and theory.