Hessian and increasing-Hessian orderings of scale-shape mixtures of multivariate skew-normal distributions and applications

Abstract

In this work, we establish some stochastic comparison results for the class of scale-shape mixtures of multivariate skew-normal (SSSN) distributions. These multivariate stochastic comparisons involve Hessian and increasing-Hessian orderings as well as many of their special cases. We show that the supermodular and increasing-supermodular orderings are equivalent to the concordance and upper orthant orderings, respectively, as well as the ordering of correlations within the underlying vectors of variables. The supermodular and increasing-supermodular orderings of log-SSSN family are also shown to be equivalent to the convex ordering and increasing-convex orderings of their geometric and harmonic averages. These results are then used to order the stability of average of relatives and aggregate value of portfolios through the order of their correlations. The aggregate claims of insurance portfolios are ordered, in the sense of stop-loss ordering, through the order of their average correlations. A multivariate stop-loss order is proposed to order the multivariate aggregate claims in SSSN family and is then shown to be equivalent to the concordance and upper orthant orderings in the sense of supermodular and increasing-supermodular orderings, respectively, based on the ordering of interdependence of claims in the portfolios. In addition, the Gini index of SSSN risks is ordered, in the sense of increasing-convex order, with respect to the strength of correlations. Finally, the results are used in analyzing some real data sets to provide illustrations for the established results and practical interpretations.