Modeling insurance data using heavy-tailed distributions is of great interest for actuaries. Probability distributions present a description of risk exposure, where the level of exposure to the risk can be determined by “key risk indicators” that usually are functions of the model. Actuaries and risk managers often use such key risk indicators to determine the degree to which their companies are subject to particular aspects of risk, which arise from changes in underlying variables such as prices of equity, interest rates, or exchange rates. The present study proposes a new heavy-tailed exponential distribution that accommodates bathtub, upside-down bathtub, decreasing, decreasing-constant, and increasing hazard rates. Actuarial measures including value at risk, tail value at risk, tail variance, and tail variance premium are derived. A computational study for these actuarial measures is conducted, proving that the proposed distribution has a heavier tail as compared with the alpha power exponential, exponentiated exponential, and exponential distributions. We adopt six estimation approaches for estimating its parameters, and assess the performance of these estimators via Monte Carlo simulations. Finally, an actuarial real data set is analyzed, proving that the proposed model can be used effectively to model insurance data as compared with fifteen competing distributions.

中文翻译：

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重尾指数分布：风险度量，估计和对精算数据的应用

对于精算师来说，使用重尾分布对保险数据进行建模非常重要。概率分布描述了风险暴露，其中风险暴露水平可以通过通常是模型功能的“关键风险指标”来确定。精算师和风险管理人员通常使用此类关键风险指标来确定其公司承受特定风险的程度，这些风险是由诸如股票价格，利率或汇率等基本变量的变化引起的。本研究提出了一种新的重尾指数分布，该分布可容纳浴缸，倒置浴缸，减小，减小常数和增加危险率。得出包括风险价值，风险尾部价值，尾部方差和尾部方差溢价在内的精算指标。对这些精算方法进行了计算研究，证明与α幂指数分布，指数分布和指数分布相比，拟议的分布具有较重的尾部。我们采用六种估计方法来估计其参数，并通过蒙特卡洛模拟来评估这些估计器的性能。最后，对精算数据进行了分析，证明与十五种竞争分布相比，所提出的模型可以有效地用于对保险数据进行建模。并通过蒙特卡洛模拟评估这些估计器的性能。最后，对精算数据进行了分析，证明与十五种竞争分布相比，所提出的模型可以有效地用于对保险数据进行建模。并通过蒙特卡洛模拟评估这些估计器的性能。最后，对精算数据进行了分析，证明与十五种竞争分布相比，所提出的模型可以有效地用于对保险数据进行建模。