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  • José María Fernández-Ponce (a1), Eva María Ortega (a2) and Franco Pellerey (a3)

Recently, Belzunce, Ortega, Pellerey, and Ruiz [3] have obtained stochastic comparisons in increasing componentwise convex order sense for vectors of random sums when the summands and number of summands depend on a common random environment, which prove how the dependence among the random environmental parameters influences the variability of vectors of random sums. The main results presented here generalize the results in Belzunce et al. [3] by considering vectors of parameters instead of a couple of parameters and the increasing directionally convex order. Results on stochastic directional convexity of families of random sums under appropriate conditions on the families of summands and number of summands are obtained, which lead to the convex comparisons between random sums mentioned earlier. Different applications in actuarial science, reliability, and population growth are also provided to illustrate the main results.

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4 C.-S. Chang , X.L. Chao , M. Pinedo & J.G. Shanthikumar (1991). Stochastic convexity for multidimensional processes and applications. IEEE Transactions on Automated Control 36: 13471355.

5 C.-S. Chang , J.G. Shanthikumar & D.D. Yao (1994). Stochastic convexity and stochastic majorization. In D.D. Yao , (ed.) Stochastic modeling and analysis of manufacturing systems New York: Springer-Verlag.

6 M. Denuit , J. Dhaene , M. Goovaerts & R. Kaas (2005). Actuarial theory for dependent risks. Measures, orders and models Chichester, UK: Wiley.

9 J.D. Esary , A.W. Marshall & F. Proschan (1973). Shock models and wear processes. The Annals of Probability 1: 627649.

13 T. Hu & L. Ruan (2004). A note on multivariate stochastic comparinsons of Bernoulli random variables. Journal of Statistical Planning and Inference 126: 281288.

15 H. Joe (1997). Multivariate models and dependence concepts London: Chapman & Hall.

16 M. Kimmel & D.E. Axelrod (2002). Branching processes in biology New York: Springer-Verlag.

17 R. Kulik (2003). Stochastic comparison of multivariate random sums. Applicationes Mathematicae 30: 379387.

18 C. Lefèvre & S. Utev (1996). Comparing sums of exchangeable Bernoulli random variables. Journal of Applied Probability 33: 285310.

19 H. Li & S. Xu (2001). Directionally convex comparison of correlated first passage times. Methodology and Computing in Applied Probability 3: 365378.

24 L.E. Meester & J.G. Shanthikumar (1999). Stochastic convexity on general space. Mathematics of Operations Research 24: 472494.

27 F. Pellerey (1993). Partial orderings under cumulative damage shock models. Advances in Applied Probability 25: 939946.

29 F. Pellerey (1999). Stochastic comparisons for multivariate shock models. Journal of Multivariate Analysis 71: 4255.

31 R.T. Rockafellar (1970). Convex analysis Princeton University Press, Princeton, NJ.

32 T. Rolski , H. Schmidli , V. Schmidt & J. Teugels (1999). Stochastic processes for insurance and finance New York: Wiley.

34 S.M. Ross & Z. Schechner (1983). Some reliability applications of the variability ordering. Operations Research 32: 679687.

35 L. Rüschendorf (2004). Comparison of multivariate risks and positive dependence. Advances in Applied Probability 41: 391406.

36 M. Shaked & J.G. Shanthikumar (1988). Stochastic convexity and its applications. Advances in Applied Probability 20: 427446.

37 M. Shaked & J.G. Shanthikumar (1988). Temporal stochastic convexity and concavity. Stochastic Processes and Their Applications 27: 120.

38 M. Shaked & J.G. Shanthikumar (1990). Parametric stochastic convexity and concavity of stochastic processes. Annals of the Institute of Statistical Mathematics 42: 509531.

39 M. Shaked & J.G. Shanthikumar (2007). Stochastic orders New-York: Springer.

40 M. Shaked & J.G. Shanthikumar (1997). Supermodular stochastic orders and positive dependence of random vectors. Journal of Multivariate Analysis 61: 86101.

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Probability in the Engineering and Informational Sciences
  • ISSN: 0269-9648
  • EISSN: 1469-8951
  • URL: /core/journals/probability-in-the-engineering-and-informational-sciences
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