Adaptive density estimation on bounded domains

Abstract

We study the estimation, in 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 socalled 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. AbstractNous étudions l'estimation, en norme 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 adapatatifs 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.where 0 < h < 1 is a bandwidth and the kernel K : R → R is such that:with 0 < γ < 1. In this context, the following lemma-which is straightforward-proves that these estimators suffer from an asymptotic pointwise bias at the endpoint 0 as soon as f (0) = 0.

Adaptive regression with Brownian path covariate

“…This specific construction allows one to obtain an estimator free of boundary bias (see Bertin, El Kolei, and Klutchnikoff, 2018, for more details). Indeed, note that for any t ∈ ∆ and under regularity assumptions on f we have:…”

“…Previous inequality follows from Taylor's expansion and classical arguments (see Bertin et al, 2018). The expression of b (k, s) can be computed:…”

Adaptive density estimation on bounded domains under mixing conditions

“…While Assumption 5 seems quite restrictive, it is satisfied by several domains as illustrated in the following examples. The first one was considered in Bertin et al (2018) for independent data.…”

“…The results stated in Müller and Stadtmüller (1999) could be used to prove pointwise minimax results over arbitrary isotropic Hölder classes. To our best knowledge only Bertin et al (2018) proved adaptive results for integrated risks over D = [0, 1] d (in the sense that a single estimation procedure achieves the minimax rate of convergence over a large scale of regularity classes). They introduced a new family of kernel density estimators that do not suffer from the boundary bias problem and they proposed a data-driven procedure based on the Goldenshluger and Lepski (see Goldenshluger and Lepski, 2014) approach that jointly selects a kernel and a bandwidth.…”

“…We assume that the observations are extracted from a stationary β-mixing process. In this framework, we introduce a new family of kernel density estimators, and we propose a data-driven selection procedure inspired by Goldenshluger and Lepski (2014) and Bertin et al (2018) to jointly select a kernel and a bandwidth. Our main contribution consists in the statement of an oracle-type inequality in L 2norm that allows us to prove the adaptivity (in the minimax sense) of our procedure over a large scale of Hölder classes without bound on the smoothness parameter.…”