Proceedings of the American Mathematical Society

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

Author(s): M. Franco; J. M. Ruiz
Journal: Proc. Amer. Math. Soc. 127 (1999), 861-874.
MSC (1991): Primary 62E10, 62G30, 60E05
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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

is a real, continuous and strictly monotonic function. We have the explicit expression of the distribution function

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.

1.: B.C. Arnold, Two characterizations of the geometric distribution, J. Appl. Probab., 17 (1980), 570-573. MR 82h:62024
2.: B.C. Arnold, N. Balakrishnan and H.N. Nagaraja, A first course in order statistics, Wiley, New York, 1992. MR 94a:62076
3.: B.C. Arnold, A. Becker, U. Gather and H. Zahedi, On the Markov property of order statistics, J. Statist. Plann. Inference, 9 (1984), 147-154. MR 85h:62011
4.: E. El-Neweihi and Z. Govindarajulu, Characterizations of geometric distribution and discrete IFR (DFR) distributions using order statistics, J. Statist. Plann. Inference, 3 (1979), 85-90. MR 81a:62012
5.: T.S. Ferguson, On characterizing distributions by properties of order statistics, Sankhy\={a} Ser. A, 29 (1967), 265-278. MR 37:2391
6.: M. Franco and J.M. Ruiz, On characterization of continuous distributions with adjacent order statistics, Statistics, 26 (1995), 375-385. CMP 96:05
7.: M. Franco and J.M. Ruiz, Characterization of discrete populations through conditional expectations of order statistics, Statist. Papers (in press).
8.: R.D. Gill and S. Johansen, A survey of product integration with a view toward application in survival analysis, Ann. Statist., 4 (1990), 1501-1555. MR 92f:60125
9.: A.H. Khan and M.S. Abu-Salih, Characterizations of probability distributions by conditional expectations of order statistics, Metron, 47 (1989), 171-181. CMP 91:10
10.: H.N. Nagaraja, On the non-markovian structure of discrete order statistics, J. Statist. Plann. Inference, 7 (1982), 29-33. MR 84a:62075
11.: H.N. Nagaraja, Some characterizations of continuous distributions based on regressions of adjacent order statistics and record values, Sankhy\={a} Ser. A, 50 (1988a), 70-73. MR 91e:62035
12.: H.N. Nagaraja, Some characterizations of discrete distributions based on linear regressions of adjacent order statistics, J. Statist. Plann. Inference, 20 (1988b), 65-75. MR 90a:62046
13.: L.Y. Ouyang, Characterizations of the uniform distribution by conditional expectations, Internat. J. Inform. Management Sci., 4 (1993), 107-111. MR 94g:62020
14.: L.Y. Ouyang, Characterization through the conditional expectation of a function of one order statistic relative to an adjacent one, Sankhy\={a} Ser. A, 57 (1995), 500-503. CMP 96:13
15.: G.S. Rogers, An alternative proof of the characterization of the density , Amer. Math. Monthly, 70 (1963), 857-858.
16.: P. Zoroa and J.M. Ruiz, El producto integral de Riemann-Stieltjes y su aplicación a las funciones de medias de las distribuciones truncadas, Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid, 79 (1985), 39-51. MR 87f:60024

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 62E10, 62G30, 60E05

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: 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
Copyright of article: Copyright 1999, American Mathematical Society

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