Adaptive kernel estimation of the mode in a nonparametric random design regression model

In a nonparametric regression model with random design, where the regression function m is given by m (x) = E(Y |X = x), estimation of the location θ (mode) and size m (θ) of a unique maximum of m is considered. As estimators, location θ and size m (θ) of a maximum of the Nadaraya-Watson kernel estimator m for the curve m are chosen. Within this setting, we establish joint asymptotic normality and asymptotic independence for θ and m (θ) (which can be exploited for constructing simultaneous confidence intervals for θ and m (θ)) under mild local smoothness assumptions on m and the design density g (imposed in a neighborhood of θ). The bandwidths employed for m are data-dependent and of plug-in type. This is handled by viewing the estimators as stochastic processes indexed by a so-called scaling parameter and proving functional central limit theorems for those processes. In the same way, we obtain, as a by-product, an asymptotic normality result for the Nadaraya-Watson estymator itself at a finite number of distinct points, which improves on previous results.