In this paper we analyze a large class of semiparametric M −estimators for singleindex 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 M −estimation problem, we propose a natural bandwidth choice by joint maximization of the M −estimation criterion with respect to the parameter of interest and the bandwidth. In this way we extend a methodology first introduced by Härdle, Hall and Ichimura (1993) 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 M −estimator when it replaces the usual bandwidth chosen by cross-validation.

International audienceFor tests based on nonparametric methods, power crucially depends on the dimension of the conditioning variables, and specifically decreases with this dimension. This is known as the “curse of dimensionality”. We propose a new general approach to nonparametric testing in high dimensional settings and we show how to implement it when testing for a parametric regression. The resulting test behaves against directional local alternatives almost as if the dimension of the regressors was one. It is also almost optimal against classes of one-dimensional alternatives for a suitable choice of the smoothing parameter. The test performs well in small samples compared to several other test