On nonparametric kernel estimation of the mode of the regression function in the random design model

In the nonparametric regression model with random design, where the regression function m is given by m(x) = E(Y \ X = x), estimation of the location theta (mode) of a unique maximum of m by the location (theta) over cap of a maximum of the Nadaraya-Watson kernel estimator (m) over cap for the curve m is considered. Within this setting, we obtain consistency and asymptotic normality results for (theta) over cap under very mild assumptions on m, the design density g of X and the kernel K. The bandwidths being considered in the present work are data-dependent of the type being generated by plug-in methods. The estimation of the size of the maximum is also considered as well as the estimation of a unique zero of the regression function. Applied to the estimation of the mode of a density, our methods yield some improvements on known results. As a by-product, we obtain some uniform consistency results for the (higher) derivatives of the Nadaraya-Watson estimator with a certain additional uniformity in the bandwiths. The proofs of those rely heavily on empirical process methods.