Hessian orderings of multivariate normal variance-mean mixture distributions and their applications in evaluating dependent multivariate risk portfolios

In this paper, some stochastic comparison results are developed for the class of multivariate normal variance-mean mixture (NVM) distributions. These comparisons are done based on Hessian and increasing Hessian orderings as well as several of their special cases. Necessary and/or sufficient conditions of the orderings are provided simply based on a comparison of the underlying model parameters. Furthermore, some linear stochastic orderings are expressed as equivalent to some multivariate orderings. The adequacy region of correlation-based ordering for the dependence structure ordering is extended from multivariate normal to NVM family by expressing supermodular order as equivalent to pairwise correlation order in NVM distributions. Some interpretations and applications of the results to actuarial science, finance and reliability are also provided. The results are finally illustrated with two real data sets.