General Marshall–Olkin Models, Dependence Orders, and Comparisons of Environmental Processes

  • Esther Frostig
  • Franco PellereyEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 141)


In many applicative fields, the behavior of a process \(\mathbf {Z}\) is assumed to be subjected to an underlying process \(\varTheta \) that describes evolutions of environmental conditions. A common way to define the environmental process is by letting the marginal values of \(\varTheta \) subjected to specific environmental factors (constant along time) and factors describing the conditions of the environment at the specified time. In this paper we describe some recent results that can be used to compare two of such environmental discrete-time processes \(\varTheta \) and \(\widetilde{\varTheta }\) in dependence. A sample of applications of the effects of these comparison results on the corresponding processes \(\mathbf {Z}\) and \(\widetilde{\mathbf {Z}}\) in some different applicative contexts are provided.


  1. 1.
    Denuit, M., Frostig, E., Levikson, B.: Supermodular comparison of time-to-ruin random vectors. Methodol. Comput. Appl. Probab. 9, 41–54 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Mari, D.D., Kotz, S.: Correlation and Dependence. Imperial College Press, London (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Fang, R., Li, X.: A note on bivariate dual generalized Marshall-Olkin distributions with applications. Probab. Eng. Inf. Sci. 27, 367–374 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Fernández-Ponce, J.M., Ortega, E., Pellerey, F.: Convex comparisons for random sums in random environments and application. Probab. Eng. Inf. Sci. 22, 389–413 (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Frostig, E.: Comparison of portfolios which depend on multivariate Bernoulli random variables with fixed marginals. Insur.: Math. Econ. 29, 319–331 (2001)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Frostig, E.: Ordering ruin probabilities for dependent claim streams. Insur.: Math. Econ. 32, 93–114 (2003)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Frostig, E., Pellerey, F.: A general framework for the supermodular comparisons of models incorporating individual and common factors. Technical Report, Department of Statistics, University of Haifa, Israel (2013)Google Scholar
  8. 8.
    Di Crescenzo, A., Frostig, E., Pellerey, F.: Stochastic comparisons of symmetric supermodular functions of heterogeneous random vectors. J. Appl. Probab. 50, 464–474 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Li, X., Pellerey, F.: Generalized Marshall-Olkin distributions and related bivariate aging properties. J. Multivar. Anal. 102, 1399–1409 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Lin, J., Li, X.: Multivariate generalized Marshall-Olkin distributions and copulas. Methodol. Comput. Appl. Probab. 16, 53–78 (2012)CrossRefGoogle Scholar
  11. 11.
    Marshall, A.W., Olkin, I.: A multivariate exponential distribution. J. Am. Stat. Assoc. 62, 30–41 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Muliere, P., Scarsini, M.: Characterization of a Marshall-Olkin type class of distributions. Ann. Inst. Stat. Math. 39, 429–441 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Müller, A.: Stop-loss order for portfolios of dependent risks. Insur.: Math. Econ. 21, 219–223 (1997)zbMATHGoogle Scholar
  14. 14.
    Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)zbMATHGoogle Scholar
  15. 15.
    Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  16. 16.
    Nyrhinen, H.: Rough descriptions of ruin for a general class of surplus processes. Adv. Appl. Probab. 30, 1008–1026 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Pellerey, F., Shaked, M., Sekeh, S.Y.: Comparisons of concordance in additive models. Stat. Probab. Lett. 582, 2059–2067 (2012)CrossRefGoogle Scholar
  18. 18.
    Ross, S.M.: Stochastic Processes. Wiley, New York (1983)zbMATHGoogle Scholar
  19. 19.
    Rüschendorf, L.: Comparison of multivariate risks and positive dependence. Adv. Appl. Probab. 41, 391–406 (2004)CrossRefzbMATHGoogle Scholar
  20. 20.
    Shaked, M., Shanthikumar, J.G.: Supermodular stochastic orders and positive dependence of random vectors. J. Multivar. Anal. 61, 86–101 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer, New York (2007)CrossRefzbMATHGoogle Scholar
  22. 22.
    Xu, S.H.: Structural analysis of a queueing system with multi-classes of correlated arrivals and blocking. Oper. Res. 47, 263–276 (1999)CrossRefGoogle Scholar
  23. 23.
    Xu, S.H., Li, H.: Majorization of weighted trees: a new tool to study correlated stochastic systems. Math. Oper. Res. 25, 298–323 (2000)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of HaifaHaifaIsrael
  2. 2.Dipartimento di Scienze MatematichePolitecnico di TorinoTurinItaly

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