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Generalized Increasing Convex and Directionally Convex Orders

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Michel M. Denuit  and
Mhamed Mesfioui
Michel M. Denuit*
Affiliation:
Université Catholique de Louvain
Mhamed Mesfioui*
Affiliation:
Université du Québec à Trois-Rivières
*
∗Postal address: Institut de Statistique and Institut des Sciences Actuarielles, Université Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium. Email address: [email protected]
∗∗Postal address: Département de Mathématiques et d'Informatique, Université du Québec à Trois-Rivières, Trois-Rivières (Québec), G9A 5H7 Canada.
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Abstract

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In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

Keywords

Integral stochastic ordersupermodular functiondirectionally convex function

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 264 - 276
Copyright
Copyright © Applied Probability Trust 2010 

References

Boutsikas, M. V. and Vaggelatou, E. (2002). On the distance between convex-ordered random variables, with applications. Adv. Appl. Prob. 34, 349–374.CrossRefGoogle Scholar
Denuit, M., Lefèvre, C. and Mesfioui, M. (1999). A class of bivariate stochastic orderings, with applications in actuarial sciences. Insurance Math. Econom. 24, 31–50.Google Scholar
Denuit, M., Lefèvre, C. and Shaked, M. (1998). The s-convex orders among real random variables, with applications. Math. Inequal. Appl. 1, 585–613.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. J. and Kaas, R. (2005). Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley, New York.Google Scholar
Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Whitt, W. (1986). Stochastic comparisons for non-Markov processes. Math. Operat. Res. 11, 609–618.Google Scholar