Generalized Location-Scale Mixtures of Elliptical Distributions: Definitions and Stochastic Comparisons

TL;DR

This paper introduces a unified class of generalized elliptical distributions with mixed location and scale parameters, providing conditions for stochastic comparisons and applications in probability and actuarial science.

Contribution

It defines a new class of generalized elliptical distributions with mixed parameters and establishes stochastic ordering conditions for these distributions.

Findings

Derived necessary and sufficient conditions for stochastic comparisons.

Provided assumptions to control tail behavior of elliptical distributions.

Illustrated applications in probability and actuarial science.

Abstract

This paper proposes a unified class of generalized location-scale mixture of multivariate elliptical distributions and studies integral stochastic orderings of random vectors following such distributions. Given a random vector $\boldsymbol{Z}$, independent of $\boldsymbol{X}$ and $\boldsymbol{Y}$, the scale parameter of this class of distributions is mixed with a function $\alpha(\boldsymbol{Z})$ and its skew parameter is mixed with another function $\beta(\boldsymbol{Z})$. Sufficient (and necessary) conditions are established for stochastically comparing different random vectors stemming from this class of distributions by means of several stochastic orders including the usual stochastic order, convex order, increasing convex order, supermodular order, and some related linear orders. Two insightful assumptions for the density generators of elliptical distributions, aiming to control the generators' tail, are provided to make stochastic comparisons among mixed-elliptical vectors. Some applications in applied probability and actuarial science are also provided as illustrations on the main findings.