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Uniqueness of characterization of absolutely continuous distributions by regressions of generalized order statistics

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  • Mariusz Bieniek

    (Maria Curie Skłodowska University)

  • Krystyna Macia̧g

    (Ernst and Young Ltd.)

Abstract

We provide a new approach to the problem of the unique identification of distributions with a continuous density by a single regression function of order statistics or record values or, more generally, generalized order statistics. Using their Markov property we show that the characterization is unique if and only if the corresponding system of differential equations has the unique solution. This result is new even in the particular case of ordinary order statistics. This approach provides a new proof of characterization of power, exponential and Pareto distributions by linearity of corresponding regression. It also yields new examples of characterizations of distributions.

Suggested Citation

  • Mariusz Bieniek & Krystyna Macia̧g, 2018. "Uniqueness of characterization of absolutely continuous distributions by regressions of generalized order statistics," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 102(3), pages 359-380, July.
    • Handle: RePEc:spr:alstar:v:102:y:2018:i:3:d:10.1007_s10182-017-0310-7
    DOI: 10.1007/s10182-017-0310-7
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    References listed on IDEAS

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    1. Erhard Cramer & Udo Kamps, 2003. "Marginal distributions of sequential and generalized order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 58(3), pages 293-310, December.
    2. Mariusz Bieniek & Dominik Szynal, 2003. "Characterizations of distributions via linearity of regression of generalized order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 58(3), pages 259-271, December.
    3. Mariusz Bieniek, 2007. "On characterizations of distributions by regression of adjacent generalized order statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 66(2), pages 233-242, September.
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