On Improving Convergence Rates for Nonnegative Kernel Density Estimators

Abstract

To improve the rate of decrease of integrated mean square error for nonparametric kernel density estimators beyond $0(n^{-\frac{4}{5}}),$ we must relax the constraint that the density estimate be a bonafide density function, that is, be nonnegative and integrate to one. All current methods for kernel (and orthogonal series) estimators relax the nonnegativity constraint. In this paper we show how to achieve similar improvement by relaxing the integral constraint only. This is important in applications involving hazard function and likelihood ratios where negative density estimates are awkward to handle.