Minimax properties of beta kernel estimators

Bertin, Karine; Klutchnikoff, Nicolas

doi:10.1016/j.jspi.2011.01.009

Cited by 17 publications

(11 citation statements)

References 4 publications

(3 reference statements)

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“…These are exploiting the results recently developed in Diel and Lerasle [2018] for the stochastic part of the risk. We are of course also using some of the results on the beta-kernel estimator presented in , Mnatsakanov [2008], Bertin and Klutchnikoff [2010]. Our rates do not match minimax rates of density estimation in i.i.d.…”

Section: Introductionmentioning

confidence: 87%

Density Estimation for RWRE

2019

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We consider the problem of non-parametric density estimation of a random environment from the observation of a single trajectory of a random walk in this environment. We first construct a density estimator using the beta-moments. We then show that the Goldenshluger-Lepski method can be used to select the beta-moment. We prove nonasymptotic bounds for the supremum norm of these estimators for both the recurrent and the transient to the right case. A simulation study supports our theoretical findings.

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Section: Introductionmentioning

confidence: 87%

Density Estimation for RWRE

2019

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“…Even if this continuously deforming seems to be an attractive property there are still some drawbacks to using such approaches. On the one hand, the beta kernels cannot be used to estimate smooth functions (see Bertin and Klutchnikoff, 2011). On the other hand, the kernels proposed by Müller and Stadtmüller (1999) are solutions of a continuous least square problem for each estimating point.…”

Section: Boundary Kernel Estimatorsmentioning

confidence: 99%

“…For bounded data, we also mention Rousseau (2010) or Autin et al (2010) that construct adaptive estimators based on Bayesian mixtures of Beta and wavelets respectively but with an extra logarithmic term factor in the rate of convergence. Additionally note also that Beta kernel density estimators are minimax only for small smoothness (see Bertin and Klutchnikoff, 2011) and consequently neither allow us to obtain such adaptive results.…”

Section: Introductionmentioning

confidence: 99%

Adaptive density estimation on bounded domains

Bertin¹,

Kolei²,

Klutchnikoff³

2019

Ann. Inst. H. Poincaré Probab. Statist.

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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.

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“…Thus, we restrict the estimation interval for the simulation study, by using the quantiles of the observations X i (see Section 4). We may have applied boundary corrections, recently developed by Karunamuni & Alberts (2005) and Bertin & Klutchnikoff (2011) for example, but this is beyond the scope of this paper.…”

Section: Case Of Knowndmentioning

confidence: 99%

Adaptive Warped Kernel Estimators

Chagny

2014

Scandinavian J Statistics

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In this work, we develop a method of adaptive non‐parametric estimation, based on ‘warped’ kernels. The aim is to estimate a real‐valued function s from a sample of random couples (X,Y). We deal with transformed data (Φ(X),Y), with Φ a one‐to‐one function, to build a collection of kernel estimators. The data‐driven bandwidth selection is performed with a method inspired by Goldenshluger and Lepski (Ann. Statist., 39, 2011, 1608). The method permits to handle various problems such as additive and multiplicative regression, conditional density estimation, hazard rate estimation based on randomly right‐censored data, and cumulative distribution function estimation from current‐status data. The interest is threefold. First, the squared‐bias/variance trade‐off is automatically realized. Next, non‐asymptotic risk bounds are derived. Lastly, the estimator is easily computed, thanks to its simple expression: a short simulation study is presented.

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Minimax properties of beta kernel estimators

Cited by 17 publications

References 4 publications

Density Estimation for RWRE

Density Estimation for RWRE

Adaptive density estimation on bounded domains

Adaptive Warped Kernel Estimators

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