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UNIFORM ORDER STATISTICS PROPERTY AND [ell ]∞-SPHERICAL DENSITIES

Published online by Cambridge University Press:  01 July 2004

Moshe Shaked ,
Fabio Spizzichino  and
Florentina Suter
Moshe Shaked
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, E-mail: [email protected]
Fabio Spizzichino
Affiliation:
Department of Mathematics, University “La Sapienza”, Piazzale Aldo Moro, 5, 00185 Rome, Italy
Florentina Suter
Affiliation:
Department of Mathematics, University of Bucharest, Academiei 14, 010014 Bucharest, Romania
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Abstract

In this article, we observe that processes with the uniform order statistics property (UOSP) can be characterized by the condition that their first n epoch times have a joint [ell ]∞≤-spherical density, n ≥ 1. Some related results, and some further properties of [ell ]∞≤-spherical densities, are also given. We also extend some of the results regarding the UOSP to the more general (not necessarily uniform) order statistics property. Finally, we develop a theory of discrete-time discrete-state processes with the UOSP, where the need to consider multiple jumps, at a single time point, arises.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 3 , July 2004 , pp. 275 - 297
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Benczur, A. (1968). On sequences of equivalent events and the compound Poisson process. Studia Scientiarum Mathematicarum Hungarica 3: 451–458.Google Scholar
Berman, S.M. (1980). Stationarity, isotropy and sphericity in [ell ]p. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 54: 21–23.Google Scholar
Crump, K.S. (1975). On point processes having an order statistic structure. Sankhya A 37: 396–404.Google Scholar
Deffner, A. & Haeusler, E. (1985). A characterization of order statistic point processes that are mixed Poisson processes and mixed sample processes simultaneously. Journal of Applied Probability 22: 314–323.Google Scholar
Diaconis, P. & Freedman, D. (1987). A dozen de Finetti-style results in search of a theory. Annales de L'Institut Henri Poincaré: Probabilités et Statistiques 23 (Suppl. 2): 397–423.Google Scholar
Feller, W. (1968). An introduction to probability theory and its applications, Vol. I, 3rd ed. New York: Wiley.
Grandell, J. (1997). Mixed Poisson processes. London: Chapman & Hall.
Hayakawa, Y. (2000). A new characterization property of mixed Poisson processes via Berman's Theorem. Journal of Applied Probability 37: 261–268.Google Scholar
Huang, W.-J. & Shoung, J.-M. (1994). On a study of some properties of point processes. Sankhya A 56: 67–76.Google Scholar
Iglesias, Z.P., Matús, F., Pereira, C.A.B., & Tanaka, N.I. (2003). On finite sequences conditionally uniform given minima and maxima. Technical Report, PUC, Santiago, Chile.
Johnson, N.L., Kotz, S., & Balakrishnan, N. (1997). Discrete multivariate distributions. New York: Wiley.
Johnson, N.L., Kotz, S., & Kemp, A.W. (1992). Univariate discrete distributions, 2nd ed. New York: Wiley.
Kamps, U. (1995). A concept of generalized order statistics. Stuttgart: Teubner.
Neuts, M.F. & Resnick, S.I. (1971). On the times of births in a linear birthprocess. Journal of the Australian Mathematical Society 12: 473–475.Google Scholar
Puri, P. (1982). On the characterization of point processes with the order statistic property without moment condition. Journal of Applied Probability 19: 39–51.Google Scholar
Shaked, M., Spizzichino, F., & Suter, F. (2002). Nonhomogeneous birth processes and [ell ]∞-spherical densities, with applications in reliability theory. Probability in the Engineering and Informational Sciences 16: 271–288.Google Scholar
Spizzichino, F. (2001). Subjective probability models for lifetimes. London: Chapman & Hall/CRC.