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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 91 open journal systems 


Asymptotic Normality in Density Support Estimation

Gérard Biau, Université Pierre et Marie Curie -- Paris VI
Benoît Cadre, ENS Cachan Bretagne
David M Mason, University of Delaware
Bruno Pelletier, Université Rennes 2


Abstract
Let $X_1,dots,X_n$ be $n$ independent observations drawn from a multivariate probability density $f$ with compact support $S_f$. This paper is devoted to the study of the estimator $hat{S}_n$ of $S_f$ defined as the union of balls centered at the $X_i$ and with common radius $r_n$. Using tools from Riemannian geometry, and under mild assumptions on $f$ and the sequence $(r_n)$, we prove a central limit theorem for $lambda (S_n Delta S_f)$, where $lambda$ denotes the Lebesgue measure on $mathbb R^d$ and $Delta$ the symmetric difference operation.


Full text: PDF

Pages: 2617-2635

Published on: December 9, 2009
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Electronic Journal of Probability. ISSN: 1083-6489