Journal of Probability and Statistics
Volume 2011 (2011), Article ID 564297, 27 pages
http://dx.doi.org/10.1155/2011/564297
Research Article

Joint Behaviour of Semirecursive Kernel Estimators of the Location and of the Size of the Mode of a Probability Density Function

Abdelkader Mokkadem,1 Mariane Pelletier,1 and Baba Thiam2

1Département de Mathématiques, Université de Versailles-Saint-Quentin, 45 avenue des Etats-Unis, 78035 Versailles Cedex, France
2EQUIPPE, Université Charles-De-Gaulle, Lille 3, Maison de la Recherche, Domaine Universitaire du Pont de Bois, BP 60149, 59653 Villeneuve d'Ascq Cedex, France

Received 24 May 2011; Accepted 24 August 2011

Academic Editor: Shein-Chung Chow

Copyright © 2011 Abdelkader Mokkadem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝜃 and 𝜇 denote the location and the size of the mode of a probability density. We study the joint convergence rates of semirecursive kernel estimators of 𝜃 and 𝜇 . We show how the estimation of the size of the mode allows measuring the relevance of the estimation of its location. We also enlighten that, beyond their computational advantage on nonrecursive estimators, the semirecursive estimators are preferable to use for the construction of confidence regions.