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Parameter recovery in two-component contamination mixtures: the L2 strategy

Author

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  • Sébastien Gadat

    (TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Jonas Kahn

    (IMT - Institut de Mathématiques de Toulouse UMR5219 - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - INSA Toulouse - Institut National des Sciences Appliquées - Toulouse - INSA - Institut National des Sciences Appliquées - UT - Université de Toulouse - UT2J - Université Toulouse - Jean Jaurès - UT - Université de Toulouse - UT3 - Université Toulouse III - Paul Sabatier - UT - Université de Toulouse - CNRS - Centre National de la Recherche Scientifique)

  • Clément Marteau

    (ICJ - Institut Camille Jordan - ECL - École Centrale de Lyon - Université de Lyon - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - INSA Lyon - Institut National des Sciences Appliquées de Lyon - Université de Lyon - INSA - Institut National des Sciences Appliquées - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique, PSPM - Probabilités, statistique, physique mathématique - ICJ - Institut Camille Jordan - ECL - École Centrale de Lyon - Université de Lyon - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon - INSA Lyon - Institut National des Sciences Appliquées de Lyon - Université de Lyon - INSA - Institut National des Sciences Appliquées - UJM - Université Jean Monnet - Saint-Étienne - CNRS - Centre National de la Recherche Scientifique)

  • Cathy Maugis

    (IMT - Institut de Mathématiques de Toulouse UMR5219 - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - INSA Toulouse - Institut National des Sciences Appliquées - Toulouse - INSA - Institut National des Sciences Appliquées - UT - Université de Toulouse - UT2J - Université Toulouse - Jean Jaurès - UT - Université de Toulouse - UT3 - Université Toulouse III - Paul Sabatier - UT - Université de Toulouse - CNRS - Centre National de la Recherche Scientifique)

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* = (1 - lambda*)phi + lambda*phi (. - mu*), where the shape phi is assumed to be known. We establish the optimal rates of convergence for the estimation of the mixture parameters (lambda*, mu*) is an element of (0, 1) x R-d. In particular, we prove that the classical parametric rate 1/ root n cannot be reached when at least one of these parameters is allowed to tend to 0 with n.

Suggested Citation

  • Sébastien Gadat & Jonas Kahn & Clément Marteau & Cathy Maugis, 2020. "Parameter recovery in two-component contamination mixtures: the L2 strategy," Post-Print hal-03328654, HAL.
    • Handle: RePEc:hal:journl:hal-03328654
    DOI: 10.1214/19-AIHP1007
    Note: View the original document on HAL open archive server: https://hal.science/hal-03328654
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    References listed on IDEAS

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    1. Cristina Butucea & Pierre Vandekerkhove, 2014. "Semiparametric Mixtures of Symmetric Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(1), pages 227-239, March.
    2. Rohit Kumar Patra & Bodhisattva Sen, 2016. "Estimation of a two-component mixture model with applications to multiple testing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(4), pages 869-893, September.
    3. Laurent Bordes & Céline Delmas & Pierre Vandekerkhove, 2006. "Semiparametric Estimation of a Two‐component Mixture Model where One Component is known," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(4), pages 733-752, December.
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    Cited by:

    1. De Castro, Y. & Gadat, Sébastien & Marteau, Clément & Maugis, Cathy, 2019. "SuperMix: Sparse Regularization for Mixture," TSE Working Papers 19-1040, Toulouse School of Economics (TSE), revised Sep 2020.

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    Keywords

    L-2 contrast; Parameter estimation; Rate of convergence; Two-component contamination mixture model;
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