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 (BX)-X-T, say eta(0)((BX)-X-T). To estimate the coefficient hector beta and the nonparametric component eta(0) we consider local polynomial fits based on kernel freighted conditional likelihoods, We then obtain an estimator of the regression function by simply replacing beta and eta(0) in eta(0)(beta(T)X) by these estimators. We deride the asymptotic distributions of these estimators and give the results of some numerical experiment. (C) Elsevier Science.