On the use of compactly supported density estimates in problems of discrimination
type: Item
instance of: scholarly article
title: On the use of compactly supported density estimates in problems of discrimination
zbMATH Open document ID: 0657.62041
DOI: 10.1016/0047-259X(87)90182-5
published in: Journal of Multivariate Analysis
publication date: 1987-01-01T00:00:00Z
review text: Dans ce remarquable travail, l'auteur considère le problème du choix de la fenêtre h pour l'estimation de densité par la méthode du noyau. Il utilise l'information de Kullback L(h) comme fonction de perte. Il montre que dans le cas d'une densité à support borné (qui tend vers zéro aux bords du support), si le noyau utilisé est à support borné, et si h est obtenu en minimisant L(h), alors il existe une borne aléatoire \(S_ n\) telle que pour \(h<S_ n\) \(L(h)=\infty\), et pour \(h>S_ n\) \(L(h)<\infty\). Il vérifie ensuite que si h est dans un voisinage de \(h_ 0\) \((h_ 0\) réalisant le minimum de \(L(\hat h))\) on a sous certaines hypothèses \(L(\hat h)=\infty.\) Puis il étudie le cas où h est choisi par la méthode de validation croisée de la vraisemblance. Notant par \(\tilde h\) la valeur choisie il trouve là encore une borne \(S*_ n\) qui vérifie \(\lim_{n\to \infty}\tilde h/S*_ n=1\) et \(P(\tilde h<S*_ n)>0\) (et donc \(P(L_ n(h)=\infty)>0)\). Enfin il donne des conditions sous lesquelles les deux méthodes sont équivalentes.
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62H30
zbMATH DE Number: 4074206
zbMATH Keywords: Kullback-Leibler loss
zbMATH Keywords: likelihood cross-validation
zbMATH Keywords: compactly supported kernels
zbMATH Keywords: bias
zbMATH Keywords: minimum loss
zbMATH Keywords: density estimation
zbMATH Keywords: discrimination information
reviewed by: José R. León
MaRDI profile type: Publication
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author: Peter Hall
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description: scientific article; zbMATH DE number 4074206