Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-25T04:01:40.371Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Comparisons of Symmetric Supermodular Functions of Heterogeneous Random Vectors

Part of: Distribution theory - Probability Operations research and management science 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: [email protected]
∗∗ Postal address: Department of Statistics, University of Haifa, Mount Carmel, Haifa 31905, Israel. Email address: [email protected]
∗∗∗ Postal address: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Balakrishnan, N., Belzunce, F., Sordo, M. A. and Suárez-Llorens, A. (2012). Increasing directionally convex orderings of random vectors having the same copula, and their use in comparing ordered data. J. Multivariate Anal. 105, 45–54.Google Scholar
Barlow, R. E. and Proschan, F. (1996). Mathematical Theory of Reliability. (Classics Appl. Math. 17). SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Bäuerle, N. (1997). Inequalities for stochastic models via supermodular orderings. Commun. Statist. Stoch. Models 13, 181–201.Google Scholar
Bäuerle, N. and Müller, A. (1998). Modeling and comparing dependencies in multivariate risk portfolios. ASTIN Bull. 28, 59–76.Google Scholar
Belzunce, F., Suárez-Llorens, A. and Sordo, M. A. (2012). Comparison of increasing directionally convex transformations of random vectors with a common copula. Insurance Math. Econom. 50, 385–390.CrossRefGoogle Scholar
Breuer, L. and Baum, D. (2005). An Introduction to Queueing Theory and Matrix-Analytic Methods. Springer, Dordrecht.Google Scholar
Cai, J. and Li, H. (2007). Dependence properties and bounds for ruin probabilities in multivariate compound risk models. J. Multivariate Anal. 98, 757–773.Google Scholar
Carter, M. (2001). Foundations of Mathematical Economics. MIT Press, Cambridge, MA.Google Scholar
Datta, M., Mirman, L. J., Morand, O. F. and Reffett, K.L. (2002). Monotone methods for Markovian equilibrium in dynamic economies. Stochastic equilibrium problems in economics and game theory. Ann. Operat. Res. 114, 117–144.CrossRefGoogle Scholar
Denuit, M., Frostig, E. and Levikson, B. (2007). Supermodular comparison of time-to-ruin random vectors. Methodology Comput. Appl. Prob. 9, 41–54.CrossRefGoogle Scholar
Di Crescenzo, A. and Pellerey, F. (2011). Improving series and parallel systems through mixtures of duplicated dependent components. Naval Res. Logistics 58, 411–418.Google Scholar
Di Crescenzo, A. and Pellerey, F. (2011). Stochastic comparisons of series and parallel systems with randomized independent components. Operat. Res. Lett. 39, 380–384.CrossRefGoogle Scholar
Gordon, W. J. and Newell, G. F. (1967). Closed queueing systems with exponential servers. Operat. Res. 15, 254–265.CrossRefGoogle Scholar
Kijima, M. and Ohnishi, M (1996). Portfolio selection problems via the bivariate characterization of stochastic dominance relations. Math. Finance 6, 237–277.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, W. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Meester, L. E. and Shanthikumar, J. G. (1999). Stochastic convexity on general space. Math. Operat. Res. 24, 472–494.Google Scholar
Müller, A. (1997). Stop-loss order for portfolios of dependent risks. Insurance Math. Econom. 21, 219–223.Google Scholar
Öner, K. B., Kiesmüller, G. P. and van Houtum, G. J. (2009). Monotonicity and supermodularity results for the Erlang loss system. Operat. Res. Lett. 37, 265–268.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar