Skip to main content
×
Home

CONVEX COMPARISONS FOR RANDOM SUMS IN RANDOM ENVIRONMENTS AND APPLICATIONS

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

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.

Copyright
References
Hide All
1Bäuerle N. & Müller A. (1998). Modeling and comparing dependencies in multivariate risk portfolios. Astin Bulletin 28: 5976.
2Bäuerle N. & Rolski T. (1998). A monotonicity result for the work-load in Markov-modulates queues. Journal of Applied Probability 35: 741747.
3Belzunce F., Ortega E.M., Pellerey F. & Ruiz J.M. (2006). Variability of total claim amounts under dependence between claims severity and number of events. Insurance: Mathematics and Economics 38: 460468.
4Chang C.-S., Chao X.L., Pinedo M. & Shanthikumar J.G. (1991). Stochastic convexity for multidimensional processes and applications. IEEE Transactions on Automated Control 36: 13471355.
5Chang C.-S., Shanthikumar J.G. & Yao D.D. (1994). Stochastic convexity and stochastic majorization. In Yao D.D., (ed.) Stochastic modeling and analysis of manufacturing systems New York: Springer-Verlag.
6Denuit M., Dhaene J., Goovaerts M. & Kaas R. (2005). Actuarial theory for dependent risks. Measures, orders and models Chichester, UK: Wiley.
7Denuit M., Genest C. & Marceau E. (2002). Criteria for the stochastic ordering of random sums, with acturial applications. Scandinavian Actuarial Journal 1: 316.
8Denuit M. & Müller A. (2002). Smooth generators of integral stochastic orders. Annals of Applied Probability 12: 11741184.
9Esary J.D., Marshall A.W. & Proschan F. (1973). Shock models and wear processes. The Annals of Probability 1: 627649.
10Frostig E. (2001). Comparison of portfolios which depend on multivariate Bernoulli random variables with fixed marginals. Insurance: Mathematics and Economics 29: 319331.
11Frostig E. (2003). Ordering ruin probabilities for dependent claim streams. Insurance: Mathematics and Economics 32: 93114.
12Frostig E. & Denuit M. (2006). Monotonicity results for portfolios with heterogeneous claims arrivals processes. Insurance: Mathematics and Economics 38: 484494.
13Hu T. & Ruan L. (2004). A note on multivariate stochastic comparinsons of Bernoulli random variables. Journal of Statistical Planning and Inference 126: 281288.
14Hu T. & Wu Z. (1999). On dependence of risks and stop-loss premiums. Insurance: Mathematics and Economics 24: 323332.
15Joe H. (1997). Multivariate models and dependence concepts London: Chapman & Hall.
16Kimmel M. & Axelrod D.E. (2002). Branching processes in biology New York: Springer-Verlag.
17Kulik R. (2003). Stochastic comparison of multivariate random sums. Applicationes Mathematicae 30: 379387.
18Lefèvre C. & Utev S. (1996). Comparing sums of exchangeable Bernoulli random variables. Journal of Applied Probability 33: 285310.
19Li H. & Xu S. (2001). Directionally convex comparison of correlated first passage times. Methodology and Computing in Applied Probability 3: 365378.
20Lillo R.E., Pellerey F., Semeraro P. & Shaked M. (2003). On the preservation of the supermodular order under multivariate claim models. Ricerche di Matematica 52: 7381.
21Lillo R.E. & Semeraro P., (2004). Stochastic bounds for discrete-time claim processes with correlated risks. Scandinavian Actuarial Journal 1: 113.
22Marshall A.W. & Olkin I. (1979). Inequalities: Theory of majorization and its Applications New York: Academic Press.
23Meester L.E. & Shanthikumar J.G. (1993). Regularity of stochastic processes. A theory based on directional convexity. Probability in the Engineering and Informational Sciences 7: 343360.
24Meester L.E. & Shanthikumar J.G. (1999). Stochastic convexity on general space. Mathematics of Operations Research 24: 472494.
25Müller A. (1997). Stop-loss order for portfolios of dependent risks. Insurance: Mathematics and Economics 21: 219223.
26Müller A. & Stoyan D. (2002). Comparison methods for stochastic models and risks, Chichester, UK: Wiley.
27Pellerey F. (1993). Partial orderings under cumulative damage shock models. Advances in Applied Probability 25: 939946.
28Pellerey F. (1997). Some new conditions for the increasing convex comparison of risks. Scandinavian Actuarial Journal 97: 3847.
29Pellerey F. (1999). Stochastic comparisons for multivariate shock models. Journal of Multivariate Analysis 71: 4255.
30Pellerey F. (2006). Comparison results for branching processes in random environments. Rapporto interno 9, Dipartimento di Matematica, Politecnico di Torino, Torino.
31Rockafellar R.T. (1970). Convex analysis Princeton University Press, Princeton, NJ.
32Rolski T., Schmidli H., Schmidt V. & Teugels J. (1999). Stochastic processes for insurance and finance New York: Wiley.
33Ross S.M. (1992). Applied probability models with optimization applications New York: Dover.
34Ross S.M. & Schechner Z. (1983). Some reliability applications of the variability ordering. Operations Research 32: 679687.
35Rüschendorf L. (2004). Comparison of multivariate risks and positive dependence. Advances in Applied Probability 41: 391406.
36Shaked M. & Shanthikumar J.G. (1988). Stochastic convexity and its applications. Advances in Applied Probability 20: 427446.
37Shaked M. & Shanthikumar J.G. (1988). Temporal stochastic convexity and concavity. Stochastic Processes and Their Applications 27: 120.
38Shaked M. & Shanthikumar J.G. (1990). Parametric stochastic convexity and concavity of stochastic processes. Annals of the Institute of Statistical Mathematics 42: 509531.
39Shaked M. & Shanthikumar J.G. (2007). Stochastic orders New-York: Springer.
40Shaked M. & Shanthikumar J.G. (1997). Supermodular stochastic orders and positive dependence of random vectors. Journal of Multivariate Analysis 61: 86101.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Probability in the Engineering and Informational Sciences
  • ISSN: 0269-9648
  • EISSN: 1469-8951
  • URL: /core/journals/probability-in-the-engineering-and-informational-sciences
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 12 *
Loading metrics...

Abstract views

Total abstract views: 81 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th November 2017. This data will be updated every 24 hours.