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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Characterization based on conditional expectations of adjacent order statistics:
A unified approach


Authors: M. Franco and J. M. Ruiz
Journal: Proc. Amer. Math. Soc. 127 (1999), 861-874
MSC (1991): Primary 62E10, 62G30, 60E05
MathSciNet review: 1610960
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04913-8
PII: S 0002-9939(99)04913-8
Keywords: Order statistics, $(n-r)$-out-of-$n$ system
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
Article copyright: © Copyright 1999 American Mathematical Society