Asymptotic Normality in Density Support Estimation

Gérard Biau (Université Pierre et Marie Curie -- Paris VI)
Benoit Cadre (ÉNS Cachan Bretagne)
David M Mason (University of Delaware)
Bruno Pelletier (Université Rennes 2)


Let $X_1,\ldots,X_n$ 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.

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Pages: 2617-2635

Publication Date: December 9, 2009

DOI: 10.1214/EJP.v14-722


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