This paper introduces a new class of distributions, named length-biased Weibull mixtures, in order to deal with heavy-tailed data encountered in quantitative risk modeling. As a generalization of the Erlang mixtures with common scale parameter, our proposed class possesses attractive modeling features such as flexibility to fit various distributional shapes and weak denseness in the class of distributions for all positive random variables. In particular, the asymptotic result shows that the length-biased Weibull mixture behaves like a Weibull-tail distribution, making it more appropriate to model heavy-tailed loss severity data. A method of statistical estimation using EM algorithm is discussed, and then applied to a simulated data set and real catastrophic losses for illustration.

中文翻译：

---

关于长度偏重的威布尔分布的混合以进行损失严重性建模

本文介绍了一种新的分布类型，称为长度偏倚的威布尔混合物，以处理定量风险建模中遇到的重尾数据。作为具有通用尺度参数的Erlang混合物的概括，我们提出的类具有吸引人的建模功能，例如，可以适应各种分布形状的灵活性以及所有正随机变量的分布类中的弱密度。尤其是，渐近结果表明，长度偏向的威布尔混合物表现得像威布尔尾巴分布，这使其更适合于建模重尾损失严重性数据。讨论了一种使用EM算法进行统计估计的方法，然后将该方法应用于模拟数据集和实际灾难性损失进行说明。