A multivariate extension of the increasing convex order to compare risks

In this paper, we propose a generalization of the increasing convex order to the multivariate setting to compare vectors of risks that accounts for both the marginal impacts and the dependence structures of the vectors. This generalization is suitable for comparing vectors with heterogeneous components and extends some well-known properties of the univariate increasing convex order. For example, comparisons of vectors with the same copula can be characterized in terms of the multivariate tail conditional expectations introduced by Cousin and Di Bernardino (2014). Moreover, if the copula reflects a particular positive dependence structure, the order among the vectors can be easily verified simply by checking the univariate increasing convex order of the marginals.