Block Empirical Likelihood Inference for Longitudinal Generalized Partially Linear Single-Index Models

TL;DR

This paper introduces a novel block empirical likelihood method for inference in longitudinal generalized partially linear single-index models, enabling likelihood-free confidence regions and improved stability in variance estimation.

Contribution

It develops a profile estimating-equation approach with spline approximation and constructs a BEL ratio statistic with a Wilks-type limit for joint inference, addressing challenges in correlated longitudinal data.

Findings

BEL ratio statistic follows a chi-square distribution

Method provides likelihood-free confidence regions

Simulation studies confirm finite-sample effectiveness

Abstract

Generalized partially linear single-index models (GPLSIMs) provide a flexible and interpretable semiparametric framework for longitudinal outcomes by combining a low-dimensional parametric component with a nonparametric index component. For repeated measurements, valid inference is challenging because within-subject correlation induces nuisance parameters and variance estimation can be unstable in semiparametric settings. We propose a profile estimating-equation approach based on spline approximation of the unknown link function and construct a subject-level block empirical likelihood (BEL) for joint inference on the parametric coefficients and the single-index direction. The resulting BEL ratio statistic enjoys a Wilks-type chi-square limit, yielding likelihood-free confidence regions without explicit sandwich variance estimation. We also discuss practical implementation, including constrained optimization for the index parameter, working-correlation choices, and bootstrap-based confidence bands for the nonparametric component. Simulation studies and an application to the epilepsy longitudinal study illustrate the finite-sample performance.