Nonparametric Recursive Estimation for Multivariate Derivative Functions by Stochastic Approximation Method

Important information concerning a multivariate data set, such as modal regions, is contained in the derivatives of the probability density or regression functions. Despite this importance, nonparametric estimation of higher order derivatives of the density or regression functions have received only relatively scant attention. The main purpose of the present work is to investigate general recursive kernel type estimators of function derivatives. We establish the central limit theorem for the proposed estimators. We discuss the optimal choice of the bandwidth by using the plug in methods. We obtain also the pointwise MDP of these estimators. Finally, we investigate the performance of the methodology for small samples through a short simulation study.