Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T01:39:03.176Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison of multivariate risks and positive dependence

Part of: Mathematical economics Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Ludger Rüschendorf
Ludger Rüschendorf*
Affiliation:
University of Freiburg
*
∗ Postal address: Department of Mathematical Stochastics, University of Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany. Email address: ruschen@stochastik.uni-freiburg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we extend some recent results on the comparison of multivariate risk vectors with respect to supermodular and related orderings. We introduce a dependence notion called the ‘weakly conditional increasing in sequence order’ that allows us to conclude that ‘more dependent’ vectors in this ordering are also comparable with respect to the supermodular ordering. At the same time, this ordering allows us to compare two risks with respect to the directionally convex order if the marginals increase convexly. We further state comparison criteria with respect to the directionally convex order for some classes of risk vectors which are modelled by functional influence factors. Finally, we discuss Fréchet bounds with respect to Δ-monotone functions when multivariate marginals are given. It turns out that, in the case of multivariate marginals, comonotone vectors no longer yield necessarily the largest risks but, in some cases, may even be vectors which minimize risk.

Keywords

Supermodular orderingFréchet boundpositive dependencerisk vector

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 91B30: Risk theory, insurance
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 391 - 406
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arjas, E., and Lehtonen, T. (1978). Approximating many server queues by means of single server queues. Math. Operat. Res. 3, 205–223.CrossRefGoogle Scholar
Bäuerle, N. (1997). Inequalities for stochastic models via supermodular orderings. Commun. Statist. Stoch. Models 13, 181–201.Google Scholar
Bäuerle, N. and Müller, A. (1998). Modeling and comparing dependencies in multivariate risk portfolios. ASTIN Bull. 28, 59–76.Google Scholar
Cambanis, S., Simons, G., and Stout, W. (1976). Inequalities for E k(X, Y) when the marginals are fixed. Z. Wahrscheinlichkeitsth. 36, 285–294.Google Scholar
Christofides, T. and Vaggelatou, E. (2004). A connection between supermodular ordering and positive/negative association. J. Multivariate Anal. 88, 138–151.Google Scholar
Dall’Aglio, G. (1972). Fréchet classes and compatibility of distribution functions. In Symposia Mathematica, Vol. 9, Academic Press, London, pp. 131–150.Google Scholar
Denuit, M., Dhaene, J., and Ribas, C. (2001). Does positive dependence between individual risks increase stop-loss premiums? Insurance Math. Econom. 28, 305–308.Google Scholar
Embrechts, P., Höing, A., and Juri, A. (2003). Using copulae to bound the value-at-risk for functions of dependent risks. Finance Stoch. 7, 145–167.Google Scholar
Fan, K., and Lorentz, G. G. (1954). An integral inequality. Amer. Math. Monthly 61, 626–631.Google Scholar
Lai, T., and Robbins, M. (1976). Maximally dependent random variables. Proc. Nat. Acad. Sci. USA 73, 286–288.Google Scholar
Meester, L., and Shantikumar, J. (1993). Regularity of stochastic processes: a theory of directional convexity. Prob. Eng. Inf. Sci. 7, 343–360.CrossRefGoogle Scholar
Meilijson, I., and Nadas, A. (1979). Convex majorization with an application to the length of critical paths. J. Appl. Prob. 16, 671–677.Google Scholar
Müller, A., and Scarsini, M. (2001). Stochastic comparison of random vectors with a common copula. Math. Operat. Res. 26, 723–740.Google Scholar
Müller, A., and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
O’Brien, G. (1975). The comparison method for stochastic processes. Ann. Prob. 3, 80–88.Google Scholar
Rüschendorf, L. (1980). Inequalities for the expectation of Δ-monotone functions. Z. Wahrscheinlichkeitsth. 54, 341–349.Google Scholar
Rüschendorf, L. (1981). Sharpness of Fréchet-bounds. Z. Wahrscheinlichkeitsth. 57, 293–302.CrossRefGoogle Scholar
Rüschendorf, L. (1981). Stochastically ordered distributions and monotonicity of the OC-function of sequential probability ratio tests. Math. Operationsforsch. Statist. 12, 327–338.Google Scholar
Rüschendorf, L. (1982). Random variables with maximum sums. Adv. Appl. Prob. 14, 623–632.Google Scholar
Rüschendorf, L. (1983). Solution of a statistical optimization problem by rearrangement methods. Metrika 30, 55–61.Google Scholar
Rüschendorf, L. (1986). Monotonicity and unbiasedness of tests via a.s. constructions. Statistics 17, 221–230.Google Scholar
Rüschendorf, L. (1991). Bounds for distributions with multivariate marginals. In Stochastic Orders and Decision under Risk (IMS Lecture Notes Monogr. Ser. 19), eds Mosler, K. and Scarsini, M., Institute of Mathematical Statistics, Hayward, CA, pp. 285–310.Google Scholar
Rüschendorf, L. (1991). Fréchet bounds and their applications. In Advances in Probability Distributions with Given Marginals (Math. Appl. 67), eds Dall’Aglio, G., Kotz, S. and Salinetti, G., Kluwer, Dordrecht, pp. 151–188.CrossRefGoogle Scholar
Rüschendorf, L. (2003). Stochastic ordering of risks, influence of dependence, and a.s. constructions. Preprint 14, University of Freiburg.Google Scholar
Shaked, M., and Tong, Y. (1985). Some partial orderings of exchangeable random variables by positive dependence. J. Multivariate Analysis 17, 333–349.Google Scholar
Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.CrossRefGoogle Scholar
Tchen, A. (1980). Inequalities for distributions with given marginals. Ann. Prob. 8, 814–827.Google Scholar