Comparison of increasing directionally convex transformations of random vectors with a common copula

Abstract

Let X and Y be two random vectors in Rn sharing the same dependence structure, that is, with a common copula. As many authors have pointed out, results of the following form are of interest: under which conditions, the stochastic comparison of the marginals of X and Y is a sufficient condition for the comparison of the expected values for some transformations of these random vectors? Assuming that the components are ordered in the univariate dispersive order–which can be interpreted as a multivariate dispersion ordering between the vectors–the main purpose of this work is to show that a weak positive dependence property, such as the positive association property, is enough for the comparison of the variance of any increasing directionally convex transformation of the vectors. Some applications in premium principles, optimization and multivariate distortions are described.