On semiparametric MM-estimation in single-index regression

In this paper we analyze a large class of semiparametric MM-estimators for single-index models, including semiparametric quasi-likelihood and semiparametric maximum likelihood estimators. Some possible applications to robustness are also mentioned. The definition of these estimators involves a kernel regression estimator for which a bandwidth rule is necessary. Given the semiparametric MM-estimation problem, we propose a natural bandwidth choice by joint maximization of the MM-estimation criterion with respect to the parameter of interest and the bandwidth. In this way we extend a methodology first introduced by Härdle et al. (Ann. Statist. 21 (1993) 157) for semiparametric least-squares. We prove asymptotic normality for our semiparametric estimator. We derive the asymptotic equivalence between our bandwidth and the optimal bandwidth obtained through weighted cross-validation. Empirical evidence obtained from simulations suggests that our bandwidth improves the higher order asymptotics of the semiparametric MM-estimator when it replaces the usual bandwidth chosen by cross-validation.