On Plug-in Rules for Local Smoothing of Density Estimators

Abstract

Optimal local smoothing of a curve estimator requires knowledge of various derivatives of the curve in the neighbourhood of the point at which estimation is being conducted. One empirical approach to selecting the amount of smoothing is to employ pilot estimators to approximate those derivatives, and substitute the approximate values into an analytical formula for the desired local bandwidth. In the present paper we study how bandwidth choice for the pilot estimators affects the performance of the final estimator. Our conclusions are rather curious. Depending on circumstance, the pilot estimators should be substantially oversmoothed or undersmoothed, relative to the amount of smoothing that would be optimal if they were to be employed themselves for point estimation. Occasionally, the optimal amount of undersmoothing is so extreme as to render the pilot estimators inconsistent. Here, the resulting local bandwidth is asymptotically random; it is not asymptotic to a sequence of constants.