In Finance and Actuarial Science, the multivariate elliptical family of distributions is a famous and well-used model for continuous risks. However, it has an essential shortcoming: all its univariate marginal distributions are the same, up to location and scale transformations. For example, all marginals of the multivariate Student’s t-distribution, an important member of the elliptical family, have the same number of degrees of freedom. We introduce a new approach to generate a multivariate distribution whose marginals are elliptical random variables, while in general, each of the risks has different elliptical distribution, which is important when dealing with insurance and financial data. The proposal is an alternative to the elliptical copula distribution where, in many cases, it is very difficult to calculate its risk measures and risk capital allocation. We study the main characteristics of the proposed model: characteristic and density functions, expectations, covariance matrices and expectation of the linear regression vector. We calculate important risk measures for the introduced distributions, such as the value at risk and tail value at risk, and the risk capital allocation of the aggregated risks.

中文翻译：

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用不同的椭圆分量对相关风险的随机向量进行建模

在金融和精算科学中，多元椭圆分布族是一种著名且广泛使用的连续风险模型。然而，它有一个本质的缺点：它的所有单变量边际分布都是相同的，直到位置和尺度变换。例如，作为椭圆家族的重要成员，多元学生 t 分布的所有边际具有相同数量的自由度。我们引入了一种新方法来生成多元分布，其边际是椭圆随机变量，而一般来说，每个风险都有不同的椭圆分布，这在处理保险和金融数据时很重要。该提议是椭圆 copula 分布的替代方案，在许多情况下，计算其风险度量和风险资本配置非常困难。我们研究了所提出模型的主要特征：特征和密度函数、期望、协方差矩阵和线性回归向量的期望。我们计算了引入分布的重要风险度量，例如风险值和风险尾值，以及聚合风险的风险资本配置。