Weak Convergence and Efficient Density Estimation at a Point

Abstract

We consider estimators for a multivariate probability density at a point. Efficient choices require knowledge of the density and its second derivatives although these are not known. We use consistent, but not necessarily efficient, estimators for these and use them to replace the unknown values in the choices for an efficient estimator. Our second stage estimators and the unattainable efficient choices are asymptotically equivalent. This follows because we show that an entire class of estimators converges weakly to a limiting stochastic process. We find asymptotically efficient estimators of kernel type.