Electronic Journal of Probability

Asymptotic Normality in Density Support Estimation

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

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Support estimation Nonparametric statistics Central limit theorem Tubular neighborhood

This work is licensed under aCreative Commons Attribution 3.0 License.


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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  • Baíllo, Amparo; Cuevas, Antonio. Image estimators based on marked bins. Statistics 40 (2006), no. 4, 277–288..
  • Baíllo, Amparo; Cuevas, Antonio; Justel, Ana. Set estimation and nonparametric detection. Canad. J. Statist. 28 (2000), no. 4, 765–782..
  • Beirlant, Jan; Györfi, László; Lugosi, Gábor. On the asymptotic normality of the $Lsb 1$- and $Lsb 2$-errors in histogram density estimation. Canad. J. Statist. 22 (1994), no. 3, 309–318..
  • Beirlant, J.; Mason, D. M. On the asymptotic normality of $Lsb p$-norms of empirical functionals. Math. Methods Statist. 4 (1995), no. 1, 1–19..
  • Biau, Gérard; Cadre, Benoît; Pelletier, Bruno. Exact rates in density support estimation. J. Multivariate Anal. 99 (2008), no. 10, 2185–2207..
  • Bredon, Glen E. Topology and geometry. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1993. xiv+557 pp. ISBN: 0-387-97926-3.
  • Cuevas, Antonio; Fraiman, Ricardo; Rodríguez-Casal, Alberto. A nonparametric approach to the estimation of lengths and surface areas. Ann. Statist. 35 (2007), no. 3, 1031–1051..
  • Cuevas, Antonio; Rodríguez-Casal, Alberto. On boundary estimation. Adv. in Appl. Probab. 36 (2004), no. 2, 340–354..
  • Devroye, Luc; Wise, Gary L. Detection of abnormal behavior via nonparametric estimation of the support. SIAM J. Appl. Math. 38 (1980), no. 3, 480–488..
  • Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques. Riemannian geometry. Third edition. Universitext. Springer-Verlag, Berlin, 2004. xvi+322 pp. ISBN: 3-540-20493-8.
  • Giné, Evarist; Mason, David M.; Zaitsev, Andrei Yu. The $Lsb 1$-norm density estimator process. Ann. Probab. 31 (2003), no. 2, 719–768..
  • Gray, Alfred. Tubes. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990. xii+283 pp. ISBN: 0-201-15676-8.
  • Hall, Peter. Random, nonuniform distribution of line segments on a circle. Stochastic Process. Appl. 18 (1984), no. 2, 239–261..
  • Hall, Peter. Three limit theorems for vacancy in multivariate coverage problems. J. Multivariate Anal. 16 (1985), no. 2, 211–236..
  • Hall, Peter. Introduction to the theory of coverage processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1988. xx+408 pp. ISBN: 0-471-85702-5.
  • Härdle, W.; Park, B. U.; Tsybakov, A. B. Estimation of non-sharp support boundaries. J. Multivariate Anal. 55 (1995), no. 2, 205–218..
  • Mason, David M.; Polonik, Wolfgang. Asymptotic normality of plug-in level set estimates. Ann. Appl. Probab. 19 (2009), no. 3, 1108–1142..
  • Molchanov, Ilya S. A limit theorem for solutions of inequalities. Scand. J. Statist. 25 (1998), no. 1, 235–242..
  • Shergin, V. V. The central limit theorem for finitely dependent random variables. Probability theory and mathematical statistics, Vol. II (Vilnius, 1989), 424–431, “Mokslas” WHERE article_id=Vilnius, 1990..