A new characterization and estimation of the zero-bias bandwidth

Abstract

It is well established that bandwidths exist that can yield an unbiased non-parametric kernel density estimate at points in particular regions (e.g. convex regions) of the underlying density. These zero-bias bandwidths have superior theoretical properties, including a 1/n convergence rate of the mean squared error. However, the explicit functional form of the zero-bias bandwidth has remained elusive. It is difficult to estimate these bandwidths and virtually impossible to achieve the higher-order rate in practice. This paper addresses these issues by taking a fundamentally different approach to the asymptotics of the kernel density estimator to derive a functional approximation to the zero-bias bandwidth. It develops a simple approximation algorithm that focuses on estimating these zero-bias bandwidths in the tails of densities where the convexity conditions favourable to the existence of the zero-bias bandwidths are more natural. The estimated bandwidths yield density estimates with mean squared error that is O(n-4/5), the same rate as the mean squared error of density estimates with other choices of local bandwidths. Simulation studies and an illustrative example with air pollution data show that these estimated zero-bias bandwidths outperform other global and local bandwidth estimators in estimating points in the tails of densities.