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Multivariate Higher-Degree Stochastic Increasing Convexity

Author

Listed:
  • Michel M. Denuit

    (Université Catholique de Louvain)

  • Mhamed Mesfioui

    (Université du Québec à Trois-Rivières)

Abstract

Building on the seminal work by Shaked and Shanthikumar (Adv Appl Probab 20:427–446, 1988a; Stoch Process Appl 27:1–20, 1988b), Denuit et al. (Eng Inf Sci 13:275–291, 1999; Methodol Comput Appl Probab 2:231–254, 2000; 2001) studied the stochastic s-increasing convexity properties of standard parametric families of distributions. However, the analysis is restricted there to a single parameter. As many standard families of distributions involve several parameters, multivariate higher-order stochastic convexity properties also deserve consideration for applications. This is precisely the topic of the present paper, devoted to stochastic $$(s_1,s_2,\ldots ,s_d)$$ ( s 1 , s 2 , … , s d ) -increasing convexity of distribution families indexed by a vector $$(\theta _1,\theta _2,\ldots ,\theta _d)$$ ( θ 1 , θ 2 , … , θ d ) of parameters. This approach accounts for possible correlation in multivariate mixture models.

Suggested Citation

  • Michel M. Denuit & Mhamed Mesfioui, 2016. "Multivariate Higher-Degree Stochastic Increasing Convexity," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1599-1623, December.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:4:d:10.1007_s10959-015-0628-6
    DOI: 10.1007/s10959-015-0628-6
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    References listed on IDEAS

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    1. Moshe Shaked & J. Shanthikumar, 1990. "Parametric stochastic convexity and concavity of stochastic processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(3), pages 509-531, September.
    2. Eva María Ortega & José Alonso, 2014. "Recent issues on stochastic directional convexity, and new results on the analysis of systems for communication, information, time scales and maintenance," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 30(4), pages 479-496, July.
    3. Denuit, Michel & Lefevre, Claude & Mesfioui, M'hamed, 1999. "On s-convex stochastic extrema for arithmetic risks," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 143-155, November.
    4. Michel Denuit & Claude Lefèvre & Moshe Shaked, 2000. "Stochastic Convexity of the Poisson Mixture Model," Methodology and Computing in Applied Probability, Springer, vol. 2(3), pages 231-254, September.
    5. Belzunce, Felix & Ortega, Eva-Maria & Pellerey, Franco & Ruiz, Jose M., 2006. "Variability of total claim amounts under dependence between claims severity and number of events," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 460-468, June.
    6. Denuit, Michel & Mesfioui, Mhamed, 2013. "Ordering Functions of Random Vectors, with Application to Partial Sums," LIDAM Reprints ISBA 2013019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. Denuit, Michel & Lefevre, Claude, 1997. "Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences," Insurance: Mathematics and Economics, Elsevier, vol. 20(3), pages 197-213, October.
    8. Denuit, Michel & Mesfioui, Mhamed, 2010. "Generalized increasing convex and directionally convex orders," LIDAM Reprints ISBA 2010029, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Michel M. Denuit & Mhamed Mesfioui, 2013. "Ordering Functions of Random Vectors, with Application to Partial Sums," Journal of Theoretical Probability, Springer, vol. 26(2), pages 474-479, June.
    10. Denuit, Michel, 2010. "Positive dependence of signals," LIDAM Discussion Papers ISBA 2010025, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Kaas, R. & Hesselager, O., 1995. "Ordering claim size distributions and mixed Poisson probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 17(2), pages 193-201, October.
    12. Denuit, Michel & Mesfioui, Mhamed, 2010. "Generalized increasing convex and directionally convex orders," LIDAM Discussion Papers ISBA 2010012, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. Denuit, Michel & Lefevre, Claude & Mesfioui, M'hamed, 1999. "A class of bivariate stochastic orderings, with applications in actuarial sciences," Insurance: Mathematics and Economics, Elsevier, vol. 24(1-2), pages 31-50, March.
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