Likelihood-Based Local Polynomial Fitting for Single-Index Models

The parametric generalized linear model assumes that the conditional distribution of a response Y given a d-dimensional covariate X belongs to an exponential family and that a known transformation of the regression function is linear in X. In this paper we relax the latter assumption by considering a nonparametric function of the linear combination βTX, say η0(βTX). To estimate the coefficient vector β and the nonparametric component η0 we consider local polynomial fits based on kernel weighted conditional likelihoods. We then obtain an estimator of the regression function by simply replacing β and η0 in η0(βTX) by these estimators. We derive the asymptotic distributions of these estimators and give the results of some numerical experiments.