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Stochastic majorization of stochastically monotone families of random variables

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Haijun Li  and
Moshe Shaked
Haijun Li*
Affiliation:
University of Arizona
Moshe Shaked*
Affiliation:
University of Arizona
*
* Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA. Research supported by AFOSR Grant AFOSR-90–0201.
* Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA. Research supported by AFOSR Grant AFOSR-90–0201.
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Abstract

Stochastic majorization is a tool that has been used in many areas of probability and statistics (such as multivariate statistical analysis, queueing theory and reliability theory) in order to obtain useful bounds and inequalities. In this paper we study the relations among several notions of stochastic majorization and stochastic convexity and obtain sufficient (and sometimes necessary) conditions which imply some of these notions. Extensions and generalizations of several results in the literature are obtained. Some examples and applications regarding stochastic comparisons of order statistics are also presented in order to illustrate the results of the paper.

Keywords

STOCHASTIC ORDERINGSEMIGROUP PROPERTIESSTOCHASTIC CONVEXITY AND CONCAVITYSTOCHASTIC SCHUR CONVEXITYLOGCONVEXITY AND LOGCONCAVITYRELIABILITY THEORYHAZARD RATE ORDERINGSUBMODULAR AND SUPERMODULAR FUNCTIONSORDER STATISTICSOPTIMAL ALLOCATIONSLIKELIHOOD RATIO ORDERING

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 4 , December 1993 , pp. 895 - 913
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

Research supported by AFOSR Grant AFOSR-90-0201.

Reproduction in whole or in part is permitted for any purpose by the United States Government.

References

Barbour, A. D., Lindvall, T. and Rogers, L. C. G. (1991) Stochastic ordering of order statistics. J. Appl. Prob. 28, 278–286.CrossRefGoogle Scholar
Boland, P. J., El-Neweihi, E. and Proschan, F. (1992) Stochastic order for redundancy allocations in series and parallel systems. Adv. Appl. Prob. 24, 161–171.CrossRefGoogle Scholar
Chang, C.-S. (1992) A new ordering for stochastic majorization: theory and applications. Adv. Appl. Prob. 24, 604–634.CrossRefGoogle Scholar
Chang, C.-S. and Yao, D. D. (1990) Rearrangements, majorization and stochastic scheduling. Technical Report, IBM Research Division, Yorktown Heights, NY 10598.Google Scholar
Karlin, S. and Proschan, F. (1960) Pólya type distributions and convolutions. Ann. Math. Statist. 31, 721–736.CrossRefGoogle Scholar
Li, H. and Shaked, M. (1993) Stochastic convexity and concavity of Markov processes. Math. Operat. Res. To appear.Google Scholar
Liyanage, L. and Shanthikumar, J. G. (1992) Allocation through stochastic Schur convexity and stochastic transposition increasingness. In Stochastic Inequalities, ed. Shaked, M. and Tong, Y. L., IMS Lecture Notes Monograph Series, Volume 22, 253–273.Google Scholar
Marshall, M. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Meester, L. and Shanthikumar, J. G. (1990) Stochastic convexity on general space, Technical Report, University of California, Berkeley.Google Scholar
Nevius, S. E., Proschan, F. and Sethuraman, J. (1977) Schur functions in statistics, II. Stochastic majorization. Ann. Statist. 5, 263–273.CrossRefGoogle Scholar
Proschan, F. and Sethuraman, J. (1976) Stochastic comparisons of order statistics from homogeneous populations, with applications in reliability. J. Multivariate Anal. 6, 608–616.CrossRefGoogle Scholar
Proschan, F. and Sethuraman, J. (1977) Schur functions in statistics, I. The preservation theorem. Ann. Statist. 5, 256–262.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1988) Stochastic convexity and its applications. Adv. Appl. Prob. 20, 427–446.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1990a) Convexity of a set of stochastically ordered random variables. Adv. Appl. Prob. 22, 160–177.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1990b) Parametric stochastic convexity and concavity of stochastic processes. Ann. Inst. Statist. Math. 42, 509–531.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1992) Optimal allocation of resources to nodes of parallel and series systems. Adv. Appl. Probab. 24, 894–914.Google Scholar
Shaked, M., Shanthikumar, J. G. and Tong, Y. L. (1991) Parametric Schur convexity and arrangement montonocity properties of partial sums. Technical Report, Department of Mathematics, University of Arizona, Tucson.Google Scholar
Shanthikumar, J. G. (1987) Stochastic majorization of random variables with proportional equilibrium rates. Adv. Appl. Prob. 19, 854–872.CrossRefGoogle Scholar