On Semiparametric estimation in Single-Index Regression

In this paper we analyze a large class of semiparametric M¡estimators for single-index models, including semiparametric quasi-likelihood and semiparametric maximumlikelihood estimators. Some possible applications to robustness are also mentioned. Thede¯nition of these estimators involves a kernel regression estimator for which a bandwidthrule is necessary. Given the semiparametric M¡estimation problem, we propose a naturalbandwidth choice by joint maximization of theM¡estimation criterion with respect to theparameter of interest and the bandwidth. In this way we extend a methodology ¯rst in-troduced by HÄardle, Hall and Ichimura (1993) for semiparametric least-squares. We proveasymptotic normality for our semiparametric estimator. We derive the asymptotic equiv-alence between our bandwidth and the optimal bandwidth obtained through weightedcross-validation. Empirical evidence obtained from simulations suggests that our band-width improves the higher order asymptotics of the semiparametric M¡estimator whenit replaces the usual bandwidth chosen by cross-validation.