On the relationship between stability of extreme order statistics and convergence of the maximum likelihood kernel density estimate
type: Item
instance of: scholarly article
title: On the relationship between stability of extreme order statistics and convergence of the maximum likelihood kernel density estimate
zbMATH Open document ID: 0701.62045
author: P. Deheuvels
author: M. Broniatowski
author: Luc Devroye
published in: The Annals of Statistics
publication date: 1989-01-01T00:00:00Z
review text: Let \(X_ 1,X_ 2,..\). be an i.i.d. sequence of random variables with density function f on \({\mathbb{R}}\). Consider kernel density estimators \[ \hat f_ n(x,h)=(nh)^{-1}\sum^{n}_{j=1}K((x-X_ j)/h) \] and choose a data driven bandwidth \(\hat h_ n\) by maximizing the cross- validated likelihood product, as suggested by \textit{J. D. F. Habbema}, \textit{J. Hermans} and \textit{Vandenbroek} [A stepwise discriminant analysis program using density estimation. G. Bruckmann (ed.), Compstat 1974, 101- 110 (1975)] and \textit{R. P. W. Duin} [IEEE Trans. Comput. C-25, 1175-1179 (1976; Zbl 0359.93035)]. \textit{P. Hall} [Biometrica 69, 383-390 (1982; Zbl 0496.62038)] showed that even for densities with bounded support, \(\hat h_ n\) behaves suboptimal but leads to consistent estimates [see also \textit{Y.-S. Chow}, \textit{S. Geman} and \textit{L.-D. Wu}, Ann. Stat. 11, 25-38 (1983; Zbl 0509.62033)]. In this paper the case of densities with unbounded support is discussed. Under some minor assumptions the main result reads essentially as follows: \(\int | \hat f_ n-f| dx\to 0\) in prob. (a.s.) iff \(X_{1,n}\) and \(X_{n,n}\) (the extremes of the sample) are stable (strongly stable), i.e. there exist sequences \(\alpha_ n\), \(\beta_ n\) s.t. \(X_{1,n}-\alpha_ n\to 0\) in prob. (a.s.) and \(X_{n,n}-\beta_ n\to 0\) in prob. (a.s.). But even then the rate of convergence may be slower than \(n^{-\epsilon}\) for any \(\epsilon >0\), as shown for certain densities in the domain of attraction of Gumbel's law.
reviewed by: Ulrich Stadtmüller
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 4149370
zbMATH Keywords: L1-consistency
zbMATH Keywords: stability of extremes
zbMATH Keywords: order statistics
zbMATH Keywords: strong stability
zbMATH Keywords: kernel density estimators
zbMATH Keywords: cross-validated likelihood product
zbMATH Keywords: densities with unbounded support
zbMATH Keywords: rate of convergence
zbMATH Keywords: domain of attraction of Gumbel's law
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176347256
OpenAlex ID: W1967158285
Wikidata QID: Q101202333
DOI: 10.1214/AOS/1176347256
description: scientific article; zbMATH DE number 4149370