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Robust parameter estimation with a small bias against heavy contamination


  • Fujisawa, Hironori
  • Eguchi, Shinto


In this paper we consider robust parameter estimation based on a certain cross entropy and divergence. The robust estimate is defined as the minimizer of the empirically estimated cross entropy. It is shown that the robust estimate can be regarded as a kind of projection from the viewpoint of a Pythagorean relation based on the divergence. This property implies that the bias caused by outliers can become sufficiently small even in the case of heavy contamination. It is seen that the asymptotic variance of the robust estimator is naturally overweighted in proportion to the ratio of contamination. One may surmise that another form of cross entropy can present the same behavior as that discussed above. It can be proved under some conditions that no cross entropy can present the same behavior except for the cross entropy considered here and its monotone transformation.

Suggested Citation

  • Fujisawa, Hironori & Eguchi, Shinto, 2008. "Robust parameter estimation with a small bias against heavy contamination," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 2053-2081, October.
  • Handle: RePEc:eee:jmvana:v:99:y:2008:i:9:p:2053-2081

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    References listed on IDEAS

    1. Miyamura, Masashi & Kano, Yutaka, 2006. "Robust Gaussian graphical modeling," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1525-1550, August.
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    Cited by:

    1. Avijit Maji & Abhik Ghosh & Ayanendranath Basu, 2016. "The logarithmic super divergence and asymptotic inference properties," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 100(1), pages 99-131, January.
    2. Shogo Kato & Shinto Eguchi, 2016. "Robust estimation of location and concentration parameters for the von Mises–Fisher distribution," Statistical Papers, Springer, vol. 57(1), pages 205-234, March.
    3. repec:eee:stapro:v:129:y:2017:i:c:p:120-130 is not listed on IDEAS
    4. repec:eee:jmvana:v:161:y:2017:i:c:p:172-190 is not listed on IDEAS
    5. Chen, Ting-Li & Fujisawa, Hironori & Huang, Su-Yun & Hwang, Chii-Ruey, 2016. "On the weak convergence and Central Limit Theorem of blurring and nonblurring processes with application to robust location estimation," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 165-184.
    6. A. Basu & A. Mandal & N. Martin & L. Pardo, 2015. "Robust tests for the equality of two normal means based on the density power divergence," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(5), pages 611-634, July.
    7. Ting-Li Chen, 2015. "On the convergence and consistency of the blurring mean-shift process," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(1), pages 157-176, February.
    8. Masashi Sugiyama & Taiji Suzuki & Takafumi Kanamori, 2012. "Density-ratio matching under the Bregman divergence: a unified framework of density-ratio estimation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(5), pages 1009-1044, October.


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