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.
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Acknowledgements
The authors are very grateful to anonymous referees for many valuable comments and remarks which were very helpful during the preparation of the final version of the paper.
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Mariusz Bieniek was supported by Polish National Science Center under Grant No. 2015/19/B/ST1/03100.
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Bieniek, M., Macia̧g, K. Uniqueness of characterization of absolutely continuous distributions by regressions of generalized order statistics. AStA Adv Stat Anal 102, 359–380 (2018). https://doi.org/10.1007/s10182-017-0310-7
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DOI: https://doi.org/10.1007/s10182-017-0310-7