Local quasi-likelihood with a parametric guide

TL;DR

This paper introduces a unified estimation approach that combines parametric guidance with nonparametric methods to improve convergence rates and incorporate prior knowledge in generalized linear models.

Contribution

It proposes a new family of parametrically-guided nonparametric estimators that outperform traditional nonparametric methods in generalized linear models.

Findings

Improved convergence rates over traditional nonparametric estimators

Numerical simulations demonstrate better performance

The method effectively incorporates prior knowledge

Abstract

Generalized linear models and the quasi-likelihood method extend the ordinary regression models to accommodate more general conditional distributions of the response. Nonparametric methods need no explicit parametric specification, and the resulting model is completely determined by the data themselves. However, nonparametric estimation schemes generally have a slower convergence rate such as the local polynomial smoothing estimation of nonparametric generalized linear models studied in Fan, Heckman and Wand [J. Amer. Statist. Assoc. 90 (1995) 141--150]. In this work, we propose a unified family of parametrically-guided nonparametric estimation schemes. This combines the merits of both parametric and nonparametric approaches and enables us to incorporate prior knowledge. Asymptotic results and numerical simulations demonstrate the improvement of our new estimation schemes over the original nonparametric counterpart.