The study and comparison of one-dimensional kernel estimators – a new approach. Part 1. Theory and methods

Department of Mathematics, Wroclaw University of Environmental and Life Sciences Grunwaldzka 53, 50-357 Wroclaw, Poland

^* Corresponding author: apm.mich@gmail.com

Abstract

In this article we compare and examine the effectiveness of different kernel density estimates for some experimental data. For a given random sample X1, X2, …, Xn we present eight kernel estimators f_n of the density function f with the Gaussian kernel and with the kernel given by Epanechnikov [1] using several methods: Silverman’s rule of thumb, the Sheather–Jones method, cross-validation methods, and other better-known plug-in methods [2–5]. To assess the effectiveness of the considered estimators and their similarity, we applied a distance measure for measurable and integrable functions [6]. All numerical calculations were performed for a set of experimental data recording groundwater level at a land reclamation facility (cf. [7–8]). The goal of the paper is to present a method that allows the study of local properties of the examined kernel estimators.