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Comparison of multivariate risks and positive dependence

Part of: Distribution theory - Probability Mathematical economics

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
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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 

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