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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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Characterization based on conditional expectations of adjacent order statistics: A unified approach
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by M. Franco and J. M. Ruiz PDF
Proc. Amer. Math. Soc. 127 (1999), 861-874 Request permission

Abstract:

In this paper, we show a unified approach to the problem of characterizing general distribution functions based on the conditional expectation between adjacent order statistics, $\xi (x)=E(h(X_{r,n})\mid X_{r+1,n}=x)$ or $\overline {\xi }(x)=E(h(X_{r+1,n})\mid X_{r,n}=x)$, where $h$ is a real, continuous and strictly monotonic function. We have the explicit expression of the distribution function $F$ from the above order mean function, $\xi$ and $\overline {\xi }$, and we give necessary and sufficient conditions so that any real function can be an order mean function. Our results generalize the results given for the discrete, absolutely continuous and continuous cases. Further, we show stability theorems for these characterizations.
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Additional Information
  • M. Franco
  • Affiliation: Departamento de Estadística e I.O., Universidad de Murcia, 30100 Murcia, Spain
  • Email: mfranco@fcu.um.es
  • J. M. Ruiz
  • Affiliation: Departamento de Estadística e I.O., Universidad de Murcia, 30100 Murcia, Spain
  • Email: jmruizgo@fcu.um.es
  • Received by editor(s): June 5, 1997
  • Additional Notes: This work was partially supported by DGES (MEC), Grant PB96-1105.
  • Communicated by: Wei Y. Loh
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 861-874
  • MSC (1991): Primary 62E10, 62G30, 60E05
  • DOI: https://doi.org/10.1090/S0002-9939-99-04913-8
  • MathSciNet review: 1610960