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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.

Suggested Citation

  • 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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    References listed on IDEAS

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    1. Györfi, L. & Vajda, I., 2002. "Asymptotic distributions for goodness-of-fit statistics in a sequence of multinomial models," Statistics & Probability Letters, Elsevier, vol. 56(1), pages 57-67, January.
    2. Ayanendranath Basu & Bruce Lindsay, 1994. "Minimum disparity estimation for continuous models: Efficiency, distributions and robustness," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 46(4), pages 683-705, December.
    3. Domingo Morales & Leandro Pardo, 2001. "Some approximations to power functions of ϕ-divergence tests in parametric models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(2), pages 249-269, December.
    4. Biau, Gérard & Devroye, Luc, 2005. "Density estimation by the penalized combinatorial method," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 196-208, May.
    5. Raúl Jiménz & Yongzhao Shao, 2001. "On robustness and efficiency of minimum divergence estimators," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(2), pages 241-248, December.
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    Citations

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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. 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.
    4. Broniatowski, Michel, 2014. "Minimum divergence estimators, maximum likelihood and exponential families," Statistics & Probability Letters, Elsevier, vol. 93(C), pages 27-33.
    5. Toma, Aida & Leoni-Aubin, Samuela, 2013. "Optimal robust M-estimators using Rényi pseudodistances," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 359-373.
    6. Chalabi, Yohan & Wuertz, Diethelm, 2012. "Portfolio optimization based on divergence measures," MPRA Paper 43332, University Library of Munich, Germany.
    7. M. D. Jiménez-Gamero & A. Batsidis & M. V. Alba-Fernández, 2016. "Fourier methods for model selection," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(1), pages 105-133, February.
    8. 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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