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

拟议的研究的主要目标包括使用重尾分布在精算科学和统计。 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，因为正常密度的尾部不够重。 重尾分布对于建模包含极端观测的数据集非常有用，例如保险应用中的大额索赔。 它们也有助于构建贝叶斯统计推断方法，该方法对离群值或冲突信息的存在具有鲁棒性。 本建议分为两部分。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密度具有大范围的尾部行为：指数，多项式和对数，这对于建模重甚至超重尾部非常有用。程序的第二部分涉及使用重尾分布进行贝叶斯推断的鲁棒性。 建立了先验和似然尾部的条件，以构造鲁棒推理，这意味着后验密度不会被冲突信息或离群值破坏。 鲁棒性的条件主要关注的先验和可能性的尾部行为，特别是它表明，需要重尾分布。该研究计划可以在统计学，精算学，金融，保险和银行业中找到许多应用。 例如，金融和精算学中的极端事件，如市场崩溃或巨额保险索赔，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.