Mixed exponential distributions are frequently used in actuarial risk modeling. Distributions obtained through mixtures allow greater flexibility in the modeling of nonlife insurance loss amounts. Several research works have studied mixed exponential distributions in univariate and multivariate settings. The present article highlights the usefulness of such distributions and lays the story of the mixing technique behind them. It also explains the underlying link between all these works. We study in detail three univariate and multivariate mixed exponential distributions defined with a discrete mixing random variable (rv). For each of these three univariate (multivariate) mixed exponential distributions and using an appropriate scaling, we identify the continuous mixing rv to which converges in distribution the discrete mixing rv and the corresponding univariate (multivariate) mixed exponential distribution. In a multivariate setting, we show that these three choices of discrete mixing distributions lead us to known Archimedean copulas constructed with continuous mixing rvs. Applications in actuarial science of these distributions are presented throughout the article highlighting their many uses and useful properties.

中文翻译：

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指数分布的单变量和多变量混合，在风险建模中的应用

混合指数分布经常用于精算风险建模。通过混合获得的分布为非人寿保险损失金额的建模提供了更大的灵活性。多项研究工作研究了单变量和多变量环境中的混合指数分布。本文重点介绍了此类分布的有用性，并讲述了它们背后的混合技术故事。它还解释了所有这些作品之间的潜在联系。我们详细研究了用离散混合随机变量 (rv) 定义的三个单变量和多变量混合指数分布。对于这三个单变量（多元）混合指数分布中的每一个，并使用适当的缩放，我们确定了离散混合 rv 和相应的单变量（多元）混合指数分布收敛到的连续混合 rv。在多变量设置中，我们展示了这三种离散混合分布的选择将我们引向了由连续混合 rvs 构造的已知阿基米德联结。整篇文章都介绍了这些分布在精算科学中的应用，重点介绍了它们的许多用途和有用的特性。