Stochastic comparisons in the scale mixture of the multivariate skew-normal family of distributions based on Hessian ordering with some applications

In this paper, we compare the random vectors from the scale-mixture of multivariate skew-normal distributions, based on Hessian orderings. The necessary and sufficient conditions for this type of ordering are studied and by considering some convex cones, the results are expressed for some important cases. In the following, the increasing Hessian ordering and necessary and sufficient conditions for some important cases are investigated. Also, the linear orderings, based on usual and convex orderings, have been discussed and it has been shown that these linear orderings in the family of multivariate scale mixture of skew-normal distributions, are of the Hessian order type. The supermodular and concordance orderings, as the important tools of dependence ordering, are obtained as equivalent to the order of correlations and using these results, the order of risk or oscillations of different portfolios in economics is explained as the order of their correlations. In the insurance context, the order of risk of aggregate claims in different portfolios is obtained as equivalent to the order of average of correlations within the portfolios. Also, in the reliability context, the order of lifetimes of parallel and series systems can be expressed in terms of correlations of system components.