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Robust parameter estimation with a small bias against heavy contamination

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  • Fujisawa, Hironori
  • Eguchi, Shinto

Abstract

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

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    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. 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.
    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. Takayuki Kawashima & Hironori Fujisawa, 2023. "Robust regression against heavy heterogeneous contamination," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(4), pages 421-442, May.
    4. Gayen, Atin & Kumar, M. Ashok, 2021. "Projection theorems and estimating equations for power-law models," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    5. M. Ekström & S. M. Mirakhmedov & S. Rao Jammalamadaka, 2020. "A class of asymptotically efficient estimators based on sample spacings," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(3), pages 617-636, September.
    6. Hung Hung & Zhi†Yu Jou & Su†Yun Huang, 2018. "Robust mislabel logistic regression without modeling mislabel probabilities," Biometrics, The International Biometric Society, vol. 74(1), pages 145-154, March.
    7. Abhijit Mandal & Beste Hamiye Beyaztas & Soutir Bandyopadhyay, 2023. "Robust density power divergence estimates for panel data models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(5), pages 773-798, October.
    8. Avijit Maji & Abhik Ghosh & Ayanendranath Basu & Leandro Pardo, 2019. "Robust statistical inference based on the C-divergence family," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1289-1322, October.
    9. Paola Stolfi & Mauro Bernardi & Davide Vergni, 2022. "Robust estimation of time-dependent precision matrix with application to the cryptocurrency market," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 8(1), pages 1-25, December.
    10. 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.
    11. Liu, Yan, 2017. "Robust parameter estimation for stationary processes by an exotic disparity from prediction problem," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 120-130.
    12. 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.
    13. 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.
    14. A. Philip Dawid & Monica Musio & Laura Ventura, 2016. "Minimum Scoring Rule Inference," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(1), pages 123-138, March.
    15. Arun Kumar Kuchibhotla & Somabha Mukherjee & Ayanendranath Basu, 2019. "Statistical inference based on bridge divergences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(3), pages 627-656, June.
    16. Mingyang Ren & Sanguo Zhang & Qingzhao Zhang, 2021. "Robust high-dimensional regression for data with anomalous responses," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(4), pages 703-736, August.
    17. Hirose, Kei & Fujisawa, Hironori & Sese, Jun, 2017. "Robust sparse Gaussian graphical modeling," Journal of Multivariate Analysis, Elsevier, vol. 161(C), pages 172-190.
    18. Yuri Goegebeur & Armelle Guillou & Jing Qin, 2023. "Robust estimation of the conditional stable tail dependence function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(2), pages 201-231, April.
    19. Aida Toma & Samuela Leoni-Aubin, 2015. "Robust Portfolio Optimization Using Pseudodistances," PLOS ONE, Public Library of Science, vol. 10(10), pages 1-26, October.

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