Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Adaptive density estimation on bounded domains

Karine Bertin, Salima El Kolei, and Nicolas Klutchnikoff

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We study the estimation, in $\mathbb{L}_{p}$-norm, of density functions defined on $[0,1]^{d}$. We construct a new family of kernel density estimators that do not suffer from the so-called boundary bias problem and we propose a data-driven procedure based on the Goldenshluger and Lepski approach that jointly selects a kernel and a bandwidth. We derive two estimators that satisfy oracle-type inequalities. They are also proved to be adaptive over a scale of anisotropic or isotropic Sobolev–Slobodetskii classes (which are particular cases of Besov or Sobolev classical classes). The main interest of the isotropic procedure is to obtain adaptive results without any restriction on the smoothness parameter.


Nous étudions l’estimation, en norme $\mathbb{L}_{p}$, d’une densité de probabilté définie sur $[0,1]^{d}$. Nous construisons une nouvelle famille d’estimateurs à noyaux qui ne sont pas biaisés au bord du domaine de définition et nous proposons une procédure de sélection simultanée d’un noyau et d’une fenêtre de lissage en adaptant la méthode développée par Goldenshluger et Lepski. Deux estimateurs différents, déduits de cette procédure générale, sont proposés et des inégalités oracles sont établies pour chacun d’eux. Ces inégalités permettent de prouver que les-dits estimateurs sont adaptatifs par rapport à des familles de classes de Sobolev–Slobodetskii anisotropes ou isotropes. Dans cette dernière situation aucune borne supérieure sur le paramètre de régularité n’est imposée.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 4 (2019), 1916-1947.

Received: 23 January 2017
Revised: 20 July 2018
Accepted: 25 September 2018
First available in Project Euclid: 8 November 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties

Multivariate kernel density estimation Bounded data Boundary bias Adaptive estimation Oracle inequality Sobolev–Slobodetskii classes


Bertin, Karine; El Kolei, Salima; Klutchnikoff, Nicolas. Adaptive density estimation on bounded domains. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 4, 1916--1947. doi:10.1214/18-AIHP938.

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