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.

翻译：本文提出了一类统一的多元椭圆分布广义位置-尺度混合，并研究了服从此类分布的随机向量的积分随机序。给定与$\boldsymbol{X}$和$\boldsymbol{Y}$独立的随机向量$\boldsymbol{Z}$，该类分布的尺度参数与函数$\alpha(\boldsymbol{Z})$混合，其偏斜参数与另一函数$\beta(\boldsymbol{Z})$混合。建立了充分（及必要）条件，用于通过多种随机序（包括通常随机序、凸序、递增凸序、超模序及相关线性序）对此类分布中不同随机向量进行随机比较。为控制椭圆分布密度生成函数的尾部特性，提出了两个富有洞察力的假设，以实现混合椭圆向量间的随机比较。最后，通过在应用概率和精算科学中的实例，对主要发现进行了说明。