Electronic Journal of Probability

Asymptotic Normality in Density Support Estimation

Gérard Biau, Benoit Cadre, David Mason, and Bruno Pelletier

Full-text: Open access

Abstract

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.

Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 91, 2617-2635.

Dates
Accepted: 9 December 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819552

Digital Object Identifier
doi:10.1214/EJP.v14-722

Mathematical Reviews number (MathSciNet)
MR2570013

Zentralblatt MATH identifier
1185.62071

Subjects
Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Keywords
Support estimation Nonparametric statistics Central limit theorem Tubular neighborhood

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Biau, Gérard; Cadre, Benoit; Mason, David; Pelletier, Bruno. Asymptotic Normality in Density Support Estimation. Electron. J. Probab. 14 (2009), paper no. 91, 2617--2635. doi:10.1214/EJP.v14-722. https://projecteuclid.org/euclid.ejp/1464819552


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