We study copula-based collective risk models when the dependence structure is defined by a Farlie-Gumbel-Morgenstern (FGM) copula. By leveraging a one-to-one correspondence between the class of FGM copulas and multivariate symmetric Bernoulli distributions, we find convenient representations for the moments and Laplace-Stieltjes transform for the aggregate random variable defined from collective risk models with FGM dependence. We examine different components of this collective risk model, aiming to have a better understanding of the impact of the assumed dependence between the frequency and severity of a claim. Relying on stochastic ordering, we analyze the impact of dependence on the aggregate claim rv $S$. Even if the FGM copula may only induce moderate dependence, we illustrate through numerical examples that the cumulative effect of dependence can generate large ranges of values for the expected value, the standard deviation, the Tail-Value-at-Risk and the entropic risk measure of aggregate claim random variables within these collective risk models. Finally, we leave the theoretical setting to investigate the collective risk model with FGM dependence with observed data.

翻译：我们研究了当相依结构由Farlie-Gumbel-Morgenstern (FGM) copula定义时，基于copula的集体风险模型。通过利用FGM copula类与多元对称伯努利分布之间的一一对应关系，我们得到了具有FGM相依性的集体风险模型中聚合随机变量的矩和Laplace-Stieltjes变换的便捷表示。我们考察了该集体风险模型的不同组成部分，旨在更好地理解索赔频率与严重性之间假设的相依性所产生的影响。基于随机序理论，我们分析了相依性对聚合索赔随机变量$S$的影响。尽管FGM copula可能仅诱导中等程度的相依性，但通过数值算例我们说明，在这些集体风险模型中，相依性的累积效应可以导致聚合索赔随机变量的期望值、标准差、尾部风险价值以及熵风险度量的取值产生较大范围的变化。最后，我们脱离理论设定，利用观测数据对具有FGM相依性的集体风险模型进行了实证研究。