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Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

Part of: Operations research and management science Distribution theory - Probability Applications

Published online by Cambridge University Press:  30 January 2018

Antonio Di Crescenzo ,
Esther Frostig  and
Franco Pellerey
Antonio Di Crescenzo*
Affiliation:
Università di Salerno
Esther Frostig*
Affiliation:
University of Haifa
Franco Pellerey*
Affiliation:
Politecnico di Torino
*
∗ Postal address: Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, n. 132, Fisciano (SA) 84084, Italy. Email address: adicrescenzo@unisa.it
∗∗ Postal address: Department of Statistics, University of Haifa, Mount Carmel, Haifa 31905, Israel. Email address: frostig@stat.haifa.ac.il
∗∗∗ Postal address: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy. Email address: franco.pellerey@polito.it
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Abstract

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Consider random vectors formed by a finite number of independent groups of independent and identically distributed random variables, where those of the last group are stochastically smaller than those of the other groups. Conditions are given such that certain functions, defined as suitable means of supermodular functions of the random variables of the vectors, are supermodular or increasing directionally convex. Comparisons based on the increasing convex order of supermodular functions of such random vectors are also investigated. Applications of the above results are then provided in risk theory, queueing theory, and reliability theory, with reference to (i) net stop-loss reinsurance premiums of portfolios from different groups of insureds, (ii) closed cyclic multiclass Gordon-Newell queueing networks, and (iii) reliability of series systems formed by units selected from different batches.

Keywords

Supermodular functiondirectionally convex functionincreasing convex orderrisks portfoliocyclic queueing networkreliabilityseries system

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62P05: Applications to actuarial sciences and financial mathematics 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 464 - 474
Copyright
© Applied Probability Trust 

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