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Parametric estimation and tests through divergences and the duality technique

  • Broniatowski, Michel
  • Keziou, Amor
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    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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    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 100 (2009)
    Issue (Month): 1 (January)
    Pages: 16-36

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    Handle: RePEc:eee:jmvana:v:100:y:2009:i:1:p:16-36
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    1. Ayanendranath Basu & Bruce Lindsay, 1994. "Minimum disparity estimation for continuous models: Efficiency, distributions and robustness," Annals of the Institute of Statistical Mathematics, Springer, vol. 46(4), pages 683-705, December.
    2. Biau, Gérard & Devroye, Luc, 2005. "Density estimation by the penalized combinatorial method," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 196-208, May.
    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, vol. 10(2), pages 249-269, December.
    4. 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.
    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, vol. 10(2), pages 241-248, December.
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