Smooth estimators of distribution and density functions

Abstract

Moving Polynomial Regression (MPR) - also called Locally Weighted Regression - is used to smooth the empirical distribution function, optimizing the L 2-norm. An estimator of the density function is obtained by differentiation. It is then shown that MPR leads to kernel estimation in both cases. Using a polynomial of degree k in the MPR, yields estimators of bias order k +1, i.e. with bias proportional to h k +1, h being the smoothing parameter. Biases and variances are computed. We compare, in the class of bias order 4, two different kernels on the basis of their bias for given equal variance. These results are illustrated by examples for both simulated and real data.