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Parametric estimation and tests through divergences and the duality technique

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  • Broniatowski, Michel
  • Keziou, Amor

Abstract

We introduce estimation and test procedures through divergence optimization for discrete or continuous parametric models. This approach is based on a new dual representation for divergences. We treat point estimation and tests for simple and composite hypotheses, extending the maximum likelihood technique. Another view of the maximum likelihood approach, for estimation and tests, is given. We prove existence and consistency of the proposed estimates. The limit laws of the estimates and test statistics (including the generalized likelihood ratio one) are given under both the null and the alternative hypotheses, and approximations of the power functions are deduced. A new procedure of construction of confidence regions, when the parameter may be a boundary value of the parameter space, is proposed. Also, a solution to the irregularity problem of the generalized likelihood ratio test pertaining to the number of components in a mixture is given, and a new test is proposed, based on [chi]2-divergence on signed finite measures and the duality technique.

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  • Broniatowski, Michel & Keziou, Amor, 2009. "Parametric estimation and tests through divergences and the duality technique," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 16-36, January.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:1:p:16-36
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    Cited by:

    1. Toma, Aida & Broniatowski, Michel, 2011. "Dual divergence estimators and tests: Robustness results," Journal of Multivariate Analysis, Elsevier, vol. 102(1), pages 20-36, January.
    2. Aharon Ben-Tal & Dick den Hertog & Anja De Waegenaere & Bertrand Melenberg & Gijs Rennen, 2013. "Robust Solutions of Optimization Problems Affected by Uncertain Probabilities," Management Science, INFORMS, vol. 59(2), pages 341-357, April.
    3. Amor Keziou & Aida Toma, 2021. "A Robust Version of the Empirical Likelihood Estimator," Mathematics, MDPI, vol. 9(8), pages 1-19, April.
    4. Subtil, Ana & de Oliveira, M. Rosário & Gonçalves, Luzia, 2012. "Conditional dependence diagnostic in the latent class model: A simulation study," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1407-1412.
    5. Ben-Tal, A. & den Hertog, D. & De Waegenaere, A.M.B. & Melenberg, B. & Rennen, G., 2011. "Robust Solutions of Optimization Problems Affected by Uncertain Probabilities," Other publications TiSEM 4d43dc51-86d9-4804-8563-9, Tilburg University, School of Economics and Management.
    6. Broniatowski, Michel, 2014. "Minimum divergence estimators, maximum likelihood and exponential families," Statistics & Probability Letters, Elsevier, vol. 93(C), pages 27-33.
    7. Toma, Aida & Leoni-Aubin, Samuela, 2013. "Optimal robust M-estimators using Rényi pseudodistances," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 359-373.
    8. V. I. Bakhtin & A. V. Lebedev, 2022. "Sup-Sums Principles for F-Divergence and a New Definition for t-Entropy," Journal of Theoretical Probability, Springer, vol. 35(1), pages 350-369, March.
    9. Diaa Al Mohamad, 2018. "Towards a better understanding of the dual representation of phi divergences," Statistical Papers, Springer, vol. 59(3), pages 1205-1253, September.
    10. Chalabi, Yohan & Wuertz, Diethelm, 2012. "Portfolio optimization based on divergence measures," MPRA Paper 43332, University Library of Munich, Germany.
    11. Aida Toma & Samuela Leoni-Aubin, 2015. "Robust Portfolio Optimization Using Pseudodistances," PLOS ONE, Public Library of Science, vol. 10(10), pages 1-26, October.
    12. Gayen, Atin & Kumar, M. Ashok, 2021. "Projection theorems and estimating equations for power-law models," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    13. Toma, Aida & Leoni-Aubin, Samuela, 2010. "Robust tests based on dual divergence estimators and saddlepoint approximations," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1143-1155, May.

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