The use of heavy-tailed distributions in actuarial science and statistics

拟议研究的主要目标包括在精算科学和统计学中使用重尾分布。统计建模通常使用正态分布。然后假设极端事件发生的概率很小，因为正态密度的尾部不够重。重尾分布对于建模包含极端观测的数据集很有用，例如保险应用中的大笔索赔。它们也有助于构建对异常值或冲突信息的存在具有健壮性的贝叶斯统计推断方法。这项建议分为两部分。第一部分涉及使用重尾密度的统计建模，特别是广义指数幂(GEP)密度族。GEP密度具有很大范围的尾部行为：指数、多项式和对数，这对建模重尾甚至超重尾很有用。程序的第二部分涉及使用重尾分布的贝叶斯推断的稳健性。为了构造稳健性推断，建立了关于先验分布和似然分布尾部的条件，这意味着后验密度不受冲突信息或离群值的破坏。稳健性条件主要涉及先验分布和似然分布的尾部行为，特别是证明了需要重尾分布。这一研究程序可以在统计学、精算学、金融、保险和银行业中找到许多应用。例如，金融和精算学中极端事件的存在，如市场崩盘或大型保险索赔，需要使用重尾分布的统计方法。

The main objective of the proposed research consists of using heavy-tailed distributions in actuarial science and statistics.  Statistical modeling is usually done using the normal distribution.  It is then assumed that the probability that extreme events occur is small, since the tails of the normal density are not sufficiently heavy.  Heavy-tailed distributions are useful for modeling data sets containing extreme observations, such as large claims in insurance applications.  They are also useful to construct Bayesian statistical inference methods which are robust to the presence of outliers or conflicting information.  This proposal is divided in two parts. The first part concerns statistical modeling using heavy-tailed densities, in particular the family of generalized exponential power (GEP) densities.  The GEP density has a large range of tail behaviour: exponential, polynomial and logarithmic, which is useful to model heavy or even super-heavy tails. The second part of the program concerns robustness in Bayesian inference using heavy-tailed distributions.  Conditions on the tails of the prior and the likelihood are established in order to construct robust inference, which means that the posterior density is not corrupted by conflicting information or outliers.  Conditions for robustness concern mainly the tail behaviour of the prior and the likelihood, in particular it is shown that heavy-tailed distributions are needed. This research program can find many applications in statistics, actuarial science, finance, insurance and banking industries.  For example, the presence of extreme events in finance and actuarial science such as a market crash or large insurance claims requires statistical methods using heavy-tailed distributions.