Estimating a density and its derivatives via the minimum distance method
type: Item
instance of: scholarly article
title: Estimating a density and its derivatives via the minimum distance method
zbMATH Open document ID: 0644.62039
DOI: 10.1007/BF00318908
publication date: 1989-01-01T00:00:00Z
review text: This paper deals with minimum distance (MD) estimators and minimum penalized distance (MPD) estimators which are based on the \(L_ p\) distance. Rates of strong consistency of MPD density estimators are established within the family of density functions which have a bounded m th derivative. For the case \(p=2\), it is also proved that the MPD density estimator achieves the optimum rate of decrease of the mean integrated square error and the \(L_ 1\) error. Estimation of derivatives of the density is considered as well. In a class parametrized by entire functions, it is proved that the rate of convergence of the MD density estimator (and its derivatives) to the unknown density (its derivatives) is of order 1/\(\sqrt{n}\) in expected \(L_ 1\) and \(L_ 2\) distances. In the same class of distributions, MD estimators of unknown density and its derivatives are proved to achieve an extraordinary rate (log log n/n)\({}^{1/2}\) of strong consistency.
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 4050763
zbMATH Keywords: minimum distance estimators
zbMATH Keywords: minimum penalized distance estimators
zbMATH Keywords: rates of strong consistency
zbMATH Keywords: density estimators
zbMATH Keywords: mean integrated square error
zbMATH Keywords: estimation of derivatives
zbMATH Keywords: rate of convergence
zbMATH Keywords: Lp-estimation
author: L. Gajek
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 4050763