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Stochastic Comparison of Random Vectors with a Common Copula

Institut für Wirtschaftstheorie und Operations Research, Universität Karlsruhe, Geb. 20.21, D-76128, Karlsruhe, Germany
Dipartimento di Scienze, Università D'Annunzio, Viale Pindaro 42, I-65127 Pescara, Italy
mueller{at}wior.uni-karlsruhe.de
scarsini{at}sci.unich.it

We consider two random vectors X and Y, such that the components of X are dominated in the convex order by the corresponding components of Y. We want to find conditions under which this implies that any positive linear combination of the components of X is dominated in the convex order by the same positive linear combination of the components of Y. This problem has a motivation in the comparison of portfolios in terms of risk. The conditions for the above dominance will concern the dependence structure of the two random vectors X and Y, namely, the two random vectors will have a common copula and will be conditionally increasing. This new concept of dependence is strictly related to the idea of conditionally increasing in sequence, but, in addition, it is invariant under permutation. We will actually prove that, under the above conditions, X will be dominated by Y in the directionally convex order, which yields as a corollary the dominance for positive linear combinations. This result will be applied to a portfolio optimization problem.

Key Words: Directionally convex order; local mean preserving spread; copula, conditionally increasing random vectors; convex ordering; portfolio optimization
History: Received: April 14, 1999; revision received: March 8, 2000;revision received: March 15, 2001;

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