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Minimax estimation of a bivariate cumulative distribution function

Author

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  • Rafał Połoczański

    (Wrocław University of Science and Technology)

  • Maciej Wilczyński

    (Wrocław University of Science and Technology)

Abstract

The problem of estimating a bivariate cumulative distribution function F under the weighted squared error loss and the weighted Cramer–von Mises loss is considered. No restrictions are imposed on the unknown function F. Estimators, which are minimax among procedures being affine transformation of the bivariate empirical distribution function, are found. Then it is proved that these procedures are minimax among all decision rules. The result for the weighted squared error loss is generalized to the case where F is assumed to be a continuous cumulative distribution function. Extensions to higher dimensions are briefly discussed.

Suggested Citation

  • Rafał Połoczański & Maciej Wilczyński, 2020. "Minimax estimation of a bivariate cumulative distribution function," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(5), pages 597-615, July.
    • Handle: RePEc:spr:metrik:v:83:y:2020:i:5:d:10.1007_s00184-019-00747-0
    DOI: 10.1007/s00184-019-00747-0
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    References listed on IDEAS

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    1. Alicja Jokiel-Rokita & Ryszard Magiera, 2007. "Minimax estimation of a cumulative distribution function by converting to a parametric problem," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 66(1), pages 61-73, July.
    2. Qiqing Yu, 1992. "Minimax invariant estimator of a continuous distribution function," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(4), pages 729-735, December.
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