The study and comparison of one-dimensional kernel estimators – a new approach. Part 2. A hydrology case study

Maciej Karczewski,Andrzej Michalski,W Zielinski,B Kazmierczak,A Michalski,L Kuchar

doi:10.1051/itmconf/20182300018

Maciej Karczewski, Andrzej Michalski + Show 4 more

Open Access

https://doi.org/10.1051/itmconf/20182300018

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Abstract

The main purpose of this article is to present the numerical consequences of selected methods of kernel estimation, using the example of empirical data from a hydrological experiment [1, 2]. In the construction of kernel estimators we used two types of kernels – Gaussian and Epanechnikov – and several methods of selecting the optimal smoothing bandwidth (see Part 1), based on various statistical and analytical conditions [3–6]. Further analysis of the properties of kernel estimators is limited to eight characteristic estimators. To assess the effectiveness of the considered estimates and their similarity, we applied the distance measure of Marczewski and Steinhaus [7]. Theoretical and numerical considerations enable the development of an algorithm for the selection of locally most effective kernel estimators.

Highlights

The results presented in this paper are an essential extension of the results of the paper [2], which considered only the Gaussian kernel K and specific window smoothing dependent upon the sample size n and some parameter from the kernel K
We consider over a dozen kernel estimators of the density function f for two kernels – the Gaussian kernel and the kernel given by Epanechnikov – using several different methods: Silverman’s rule of thumb, the Sheather–Jones method, cross-validation methods and other selected plug-in methods
In the statistical literature there can be found a wealth of different approaches to obtaining the best estimate of the unknown density function of a continuous type random variable by nonparametric kernel estimation methods

Summary

Introduction

The results presented in this paper are an essential extension of the results of the paper [2], which considered only the Gaussian kernel K and specific window smoothing dependent upon the sample size n and some parameter from the kernel K. There the estimator of the unknown function f was expressed as. We consider over a dozen kernel estimators of the density function f for two kernels – the Gaussian kernel and the kernel given by Epanechnikov – using several different methods: Silverman’s rule of thumb, the Sheather–Jones method, cross-validation methods and other selected plug-in methods. The goal of this paper is to present a method that allows the study of local properties of the examined kernel estimators

Numerical results and discussion

Conclusions