Journal of Integral Equations and Applications

On the unique characterization of continuous distributions by single regression of non-adjacent generalized order statistics

Mariusz Bieniek and Krystyna Maciag

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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.

Article information

Source
J. Integral Equations Applications, Volume 30, Number 4 (2018), 491-519.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.jiea/1543482174

Digital Object Identifier
doi:10.1216/JIE-2018-30-4-491

Mathematical Reviews number (MathSciNet)
MR3881214

Zentralblatt MATH identifier
06989830

Subjects
Primary: 60E05: Distributions: general theory 62E10: Characterization and structure theory 62G30: Order statistics; empirical distribution functions

Keywords
Characterization generalized order statistics regression order statistics uniqueness of solutions to integral equations

Citation

Bieniek, Mariusz; Maciag, Krystyna. On the unique characterization of continuous distributions by single regression of non-adjacent generalized order statistics. J. Integral Equations Applications 30 (2018), no. 4, 491--519. doi:10.1216/JIE-2018-30-4-491. https://projecteuclid.org/euclid.jiea/1543482174


Export citation

References

  • N. Balakrishnan and E. Cramer, The art of progressive censoring, Stat. Indust. Tech. (2014), Birkhäuser/Springer, New York.
  • M. Bieniek, On characterizations of distributions by regression of adjacent generalized order statistics, Metrika 66 (2007), 233–242.
  • ––––, On characterizations of distributions by regression of generalized order statistics, Australian New Zealand J. Statis. 51 (2009), 89–99.
  • 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.
  • M. Bieniek and D. Szynal, Characterizations of distributions via linearity of regression of generalized order statistics, Metrika 58 (2003), 259–271.
  • E. Cramer and U. Kamps, Marginal distributions of sequential and generalized order statistics, Metrika 58 (2003), 293–310.
  • 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.
  • 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.
  • A. Dembińska and J. Wesołowski, Linearity of regression for non-adjacent order statistics, Metrika 48 (1998), 215–222.
  • ––––, Linearity of regression for non-adjacent record values, J. Stat. Plan. Infer. 90 (2000), 195–205.
  • T.S. Ferguson, On characterizing distributions by properties of order statistics, Sankhyā 29 (1967), 265–278.
  • M. Franco and J.M. Ruiz, On characterization of continuous distributions with adjacent order statistics, Statistics 26 (1995), 375–385.
  • ––––, On characterization of continuous distributions by conditional expectation of record values, Sankhyā 58 (1996), 135–141.
  • ––––, Characterization based on conditional expectations of adjacent order statistics: a unified approach, Proc. Amer. Math. Soc. 127 (1999), 861–874.
  • W.J. Kaczor and M.T. Nowak, Problems in mathematical analysis, III, Student Math. Libr. 21 (2003), American Mathematical Society, Providence, RI.
  • U. Kamps, A concept of generalized order statistics, J. Stat. Plan. Infer. 48 (1995), 1–23.
  • A.M. Mathai, A handbook of generalized special functions for statistical and physical sciences, The Clarendon Press, Oxford University Press, New York, 1993.
  • H.N. Nagaraja, On the expected values of record values, Austral. J. Statist. 20 (1978), 176–182.
  • W. Rudin, Real and complex analysis, McGraw-Hill Book Co., New York, 1987.