Empirical-Bias Bandwidths for Local Polynomial Nonparametric Regression and Density Estimation

Abstract

A data-based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials. The estimator, called the empirical-bias bandwidth selector (EBBS), is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function. EBBS minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term. The variance term used is exact, not asymptotic, though it involves the conditional variance of the response given the predictors that must be estimated. The bias term is estimated empirically, not from an asymptotic expression. Thus EBBS is similar to the “double smoothing” approach of Härdle, Hall, and Marron and a local bandwidth selector of Schucany, but is developed here for a far wider class of estimation problems than what those authors considered. EBBS is tested on simulated data, and its performance seems quite satisfactory. Local polynomial smoothing of a histogram is a highly effective technique for density estimation, and several of the examples involve density estimation by EBBS applied to binned data.