Local likelihood method: a bridge over parametric and nonparametric regression

Authors: Shinto Eguchi¹; Tae Yoon Kim²; Byeong Park³

Source: Journal of Nonparametric Statistics, Volume 15, Number 6, December 2003 , pp. 665-683(19)

Publisher: Taylor and Francis Ltd

Abstract:

This paper discusses local likelihood method for estimating a regression function in a setting which includes generalized linear models. The local likelihood function is constructed by first considering a parametric model for the regression function. It is defined as a locally weighted log-likelihood with weights determined by a kernel function and a bandwidth. When a large bandwidth is chosen, the resulting estimator would be close to the fully parametric maximum likelihood estimator, so that a large bandwidth would be a relevant choice in the case where the true regression function is near the parametric family. On the other hand, when a small bandwidth is chosen, the performance of the resulting estimator would not depend much on the assumed parametric model, thus a small bandwidth would be desirable if the parametric model is largely misspecified. In this paper, we detail the way in which the risk of the local likelihood estimator is affected by bandwidth selection and model misspecification. We derive explicit formulas for the bias and variance of the local likelihood estimator for both large and small bandwidths. We look into higher order asymptotic expansions for the risk of the local likelihood estimator in the case where the bandwidth is large, which enables us to determine the optimal size of the bandwidth depending on the degree of model misspecification.

Keywords: Local likelihood; Near-parametric model; Near-nonparametric model; Deviance function; Generalized linear model

Document Type: Research article

DOI: http://dx.doi.org/10.1080/10485250310001624756

Affiliations: 1: Institute of Statistical Mathematics Tokyo Japan 2: Department of Statistics Keimyung University Taegu South Korea 3: Department of Statistics Seoul National University Seoul South Korea

Publication date: 2003-12-01

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