On the convergence of kernel estimators of probability density functions

Abstract

The properties of the characteristic function of the fixed-bandwidth kernel estimator of a probability density function are used to derive estimates of the rate of almost sure convergence of such estimators in a family of Hilbert spaces. The convergence of these estimators in a reproducing-kernel Hilbert space is used to prove the uniform convergence of variable-bandwidth estimators. An algorithm employing the fast Fourier transform and heuristic estimates of the optimal bandwidth is presented, and numerical experiments using four density functions are described.