×

zbMATH — the first resource for mathematics

On the unique characterization of continuous distributions by single regression of non-adjacent generalized order statistics. (English) Zbl 1403.60019
Summary: We show a new and unexpected application of integral equations and their systems to the problem of the unique identification of continuous probability distributions based on the knowledge of exactly one regression function of ordered statistical data. The most popular example of such data are the order statistics which are obtained by non-decreasing ordering of elements of the sample according to their magnitude. However, our considerations are conducted in the abstract setting of so-called generalized order statistics. This model includes order statistics and other interesting models of ordered random variables. We prove that the uniqueness of characterization is equivalent to the uniqueness of the solution to the appropriate system of integral equations with non-classical initial conditions. This criterion for uniqueness is then applied to give new examples of characterizations.
MSC:
60E05 Probability distributions: general theory
62E10 Characterization and structure theory of statistical distributions
62G30 Order statistics; empirical distribution functions
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] N. Balakrishnan and E. Cramer, The art of progressive censoring, Stat. Indust. Tech. (2014), Birkhäuser/Springer, New York. · Zbl 1365.62001
[2] M. Bieniek, On characterizations of distributions by regression of adjacent generalized order statistics, Metrika 66 (2007), 233–242. · Zbl 1433.62121
[3] ——–, On characterizations of distributions by regression of generalized order statistics, Australian New Zealand J. Statis. 51 (2009), 89–99. · Zbl 1334.62028
[4] M. Bieniek and K. Maciąg, Uniqueness of characterization of absolutely continuous distributions by regressions of generalized order statistics, AStA Adv. Stat. Anal. 102 (2018), 359–380.
[5] M. Bieniek and D. Szynal, Characterizations of distributions via linearity of regression of generalized order statistics, Metrika 58 (2003), 259–271. · Zbl 1042.62004
[6] E. Cramer and U. Kamps, Marginal distributions of sequential and generalized order statistics, Metrika 58 (2003), 293–310. · Zbl 1042.62048
[7] E. Cramer, U. Kamps and C. Keseling, Characterizations via linear regression of ordered random variables: A unifying approach, Comm. Statist. Th. Meth. 33 (2004), 2885–2911. · Zbl 1087.62010
[8] E. Cramer and T.-T.-H. Tran, Generalized order statistics from arbitrary distributions and the Markov chain property, J. Statist. Plan. Infer. 139 (2009), 4064–4071. · Zbl 1183.62082
[9] A. Dembińska and J. Wesołowski, Linearity of regression for non-adjacent order statistics, Metrika 48 (1998), 215–222. · Zbl 1093.62513
[10] ——–, Linearity of regression for non-adjacent record values, J. Stat. Plan. Infer. 90 (2000), 195–205. · Zbl 0991.62005
[11] T.S. Ferguson, On characterizing distributions by properties of order statistics, Sankhyā 29 (1967), 265–278. · Zbl 0155.27302
[12] M. Franco and J.M. Ruiz, On characterization of continuous distributions with adjacent order statistics, Statistics 26 (1995), 375–385. · Zbl 0836.62013
[13] ——–, On characterization of continuous distributions by conditional expectation of record values, Sankhyā 58 (1996), 135–141. · Zbl 0885.62008
[14] ——–, Characterization based on conditional expectations of adjacent order statistics: a unified approach, Proc. Amer. Math. Soc. 127 (1999), 861–874. · Zbl 0926.62004
[15] W.J. Kaczor and M.T. Nowak, Problems in mathematical analysis, III, Student Math. Libr. 21 (2003), American Mathematical Society, Providence, RI. · Zbl 1054.00004
[16] U. Kamps, A concept of generalized order statistics, J. Stat. Plan. Infer. 48 (1995), 1–23. · Zbl 0851.62035
[17] A.M. Mathai, A handbook of generalized special functions for statistical and physical sciences, The Clarendon Press, Oxford University Press, New York, 1993. · Zbl 0770.33001
[18] H.N. Nagaraja, On the expected values of record values, Austral. J. Statist. 20 (1978), 176–182. · Zbl 0407.62025
[19] W. Rudin, Real and complex analysis, McGraw-Hill Book Co., New York, 1987. · Zbl 0925.00005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.