Local likelihood method: A bridge over parametric and nonparametric regression

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.