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Some Remarks on the Supermodular Order

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  • Müller, Alfred
  • Scarsini, Marco

Abstract

In this paper we solve two open problems posed by Joe (1997) concerning the supermodular order. First we give an example which shows that the supermodular order is strictly stronger than the concordance order for dimension d=3. Second we show that the supermodular order fulfils all desirable properties of a multivariate positive dependence order. We especially prove the non-trivial fact that it is closed with respect to weak convergence. This is applied to give a complete characterization of the supermodular order for multivariate normal distributions.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 73 (2000)
Issue (Month): 1 (April)
Pages: 107-119

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Handle: RePEc:eee:jmvana:v:73:y:2000:i:1:p:107-119

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Related research

Keywords: dependence orders; supermodular order; L-superadditive functions; weak convergence; concordance;

References

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  1. Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
  2. Shaked, Moshe & Shanthikumar, J. George, 1997. "Supermodular Stochastic Orders and Positive Dependence of Random Vectors," Journal of Multivariate Analysis, Elsevier, vol. 61(1), pages 86-101, April.
  3. Muller, Alfred, 1997. "Stop-loss order for portfolios of dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 21(3), pages 219-223, December.
  4. Block, Henry W. & Sampson, Allan R., 1988. "Conditionally ordered distributions," Journal of Multivariate Analysis, Elsevier, vol. 27(1), pages 91-104, October.
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Citations

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Cited by:
  1. Laurence, Peter & Wang, Tai-Ho, 2009. "Sharp distribution free lower bounds for spread options and the corresponding optimal subreplicating portfolios," Insurance: Mathematics and Economics, Elsevier, vol. 44(1), pages 35-47, February.
  2. Cousin, Areski & Laurent, Jean-Paul, 2008. "Comparison results for exchangeable credit risk portfolios," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 1118-1127, June.
  3. Escudero, Laureano F. & Ortega, Eva-María, 2008. "Actuarial comparisons for aggregate claims with randomly right-truncated claims," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 255-262, October.
  4. Christofides, Tasos C. & Vaggelatou, Eutichia, 2004. "A connection between supermodular ordering and positive/negative association," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 138-151, January.
  5. Marta_Cardin & Paola_Ferretti, 2004. "Some theory of bivariate risk attitude," Game Theory and Information 0411009, EconWPA.
  6. Arlotto, Alessandro & Scarsini, Marco, 2009. "Hessian orders and multinormal distributions," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2324-2330, November.
  7. 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.
  8. Bauerle, Nicole, 2002. "Risk management in credit risk portfolios with correlated assets," Insurance: Mathematics and Economics, Elsevier, vol. 30(2), pages 187-198, April.
  9. Hobson, David & Laurence, Peter & Wang, Tai-Ho, 2005. "Static-arbitrage optimal subreplicating strategies for basket options," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 553-572, December.
  10. Hu, Taizhong & Pan, Xiaoming, 1999. "Preservation of multivariate dependence under multivariate claim models," Insurance: Mathematics and Economics, Elsevier, vol. 25(2), pages 171-179, November.
  11. Hu, Taizhong & Xie, Chaode & Ruan, Lingyan, 2005. "Dependence structures of multivariate Bernoulli random vectors," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 172-195, May.
  12. Colangelo, Antonio & Scarsini, Marco & Shaked, Moshe, 2005. "Some notions of multivariate positive dependence," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 13-26, August.
  13. Muller, Alfred & Pflug, Georg, 2001. "Asymptotic ruin probabilities for risk processes with dependent increments," Insurance: Mathematics and Economics, Elsevier, vol. 28(3), pages 381-392, June.
  14. Laureano Escudero & Eva-María Ortega, 2009. "How retention levels influence the variability of the total risk under reinsurance," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 17(1), pages 139-157, July.
  15. Kulik, Rafal & Szekli, Ryszard, 2005. "Dependence orderings for some functionals of multivariate point processes," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 145-173, January.
  16. Alfred Müller, 2001. "Stochastic Ordering of Multivariate Normal Distributions," Annals of the Institute of Statistical Mathematics, Springer, vol. 53(3), pages 567-575, September.
  17. Cheung, Ka Chun & Dhaene, Jan & Kukush, Alexander & Linders, Daniël, 2013. "Ordered random vectors and equality in distribution," Open Access publications from Katholieke Universiteit Leuven urn:hdl:123456789/375156, Katholieke Universiteit Leuven.
  18. Marta Cardin & Elisa Pagani, 2008. "Some proposals about multivariate risk measurement," Working Papers 165, Department of Applied Mathematics, Università Ca' Foscari Venezia.
  19. Hennessy, David A. & Saak, Alexander E. & Babcock, Bruce A., 2003. "Fair Value Of Whole-Farm And Crop-Specific Revenue Insurance," 2003 Annual meeting, July 27-30, Montreal, Canada 21988, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).
  20. Huang, Di & Zhou, Hong & Zhao, Qiu-Hong, 2011. "A competitive multiple-product newsboy problem with partial product substitution," Omega, Elsevier, vol. 39(3), pages 302-312, June.

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