Estimating Smoothness and Optimal Bandwidth for Probability Density Functions

Dimitris N. Politis (), Peter F. Tarassenko and Vyacheslav A. Vasiliev
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Dimitris N. Politis: Department of Mathematics and Halicioglu Data Science Institute, University of California, San Diego, CA 92093-0112, USA
Peter F. Tarassenko: Institute of Applied Mathematics and Computer Science, Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia
Vyacheslav A. Vasiliev: Institute of Applied Mathematics and Computer Science, Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia

Stats, 2022, vol. 6, issue 1, 1-20

Abstract: The properties of non-parametric kernel estimators for probability density function from two special classes are investigated. Each class is parametrized with distribution smoothness parameter. One of the classes was introduced by Rosenblatt, another one is introduced in this paper. For the case of the known smoothness parameter, the rates of mean square convergence of optimal (on the bandwidth) density estimators are found. For the case of unknown smoothness parameter, the estimation procedure of the parameter is developed and almost surely convergency is proved. The convergence rates in the almost sure sense of these estimators are obtained. Adaptive estimators of densities from the given class on the basis of the constructed smoothness parameter estimators are presented. It is shown in examples how parameters of the adaptive density estimation procedures can be chosen. Non-asymptotic and asymptotic properties of these estimators are investigated. Specifically, the upper bounds for the mean square error of the adaptive density estimators for a fixed sample size are found and their strong consistency is proved. The convergence of these estimators in the almost sure sense is established. Simulation results illustrate the realization of the asymptotic behavior when the sample size grows large.

Keywords: non-parametric kernel density estimators; adaptive density estimators; mean square and almost surely convergence; rate of convergence; smoothness class (search for similar items in EconPapers)
JEL-codes: C1 C10 C11 C14 C15 C16 (search for similar items in EconPapers)
Date: 2022
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