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May 2020 Parameter recovery in two-component contamination mixtures: The $L^{2}$ strategy
Sébastien Gadat, Jonas Kahn, Clément Marteau, Cathy Maugis-Rabusseau
Ann. Inst. H. Poincaré Probab. Statist. 56(2): 1391-1418 (May 2020). DOI: 10.1214/19-AIHP1007

Abstract

In this paper, we consider a parametric density contamination model. We work with a sample of i.i.d. data with a common density, $f^{\star }=(1-\lambda^{\star })\phi +\lambda^{\star }\phi (\cdot-\mu^{\star })$, where the shape $\phi $ is assumed to be known. We establish the optimal rates of convergence for the estimation of the mixture parameters $(\lambda^{\star },\mu^{\star })\in (0,1)\times \mathbb{R}^{d}$. In particular, we prove that the classical parametric rate $1/\sqrt{n}$ cannot be reached when at least one of these parameters is allowed to tend to $0$ with $n$.

Dans cet article, nous étudions un modèle de contamination paramétrique. Nous considérons un échantillon i.i.d de densité $f^{\star }=(1-\lambda^{\star })\phi +\lambda^{\star }\phi (\cdot-\mu^{\star })$, où la fonction $\phi $ est supposée connue. Nous établissons des vitesses de convergence optimales pour l’estimation des paramètres de mélange $(\lambda^{\star },\mu^{\star })\in (0,1)\times \mathbb{R}^{d}$. En particulier, nous prouvons que la vitesse paramétrique usuelle $1/\sqrt{n}$ ne peut pas être atteinte quand au moins un de ces paramètres est amené à tendre vers $0$ avec $n$.

Citation

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Sébastien Gadat. Jonas Kahn. Clément Marteau. Cathy Maugis-Rabusseau. "Parameter recovery in two-component contamination mixtures: The $L^{2}$ strategy." Ann. Inst. H. Poincaré Probab. Statist. 56 (2) 1391 - 1418, May 2020. https://doi.org/10.1214/19-AIHP1007

Information

Received: 20 February 2018; Revised: 21 November 2018; Accepted: 27 May 2019; Published: May 2020
First available in Project Euclid: 16 March 2020

zbMATH: 07199902
MathSciNet: MR4076788
Digital Object Identifier: 10.1214/19-AIHP1007

Subjects:
Primary: 62F15, 62G05
Secondary: 62G20

Rights: Copyright © 2020 Institut Henri Poincaré

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Vol.56 • No. 2 • May 2020
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