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

重尾分布在精算科学和统计学中的应用

• 批准号: 327070-2006
• 负责人: Desgagné, Alain
• 金额: $ 0.73万
• 依托单位: Université du Québec à Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2006
• 资助国家: 加拿大
• 起止时间: 2006-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=334382
• 关键词: use; heavy; tailed; distributions; actuarial

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.

本研究的主要目的是在精算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，如保险应用中的大额索赔。它们也可用于构造贝叶斯统计推断方法，该方法对异常值或冲突信息的存在具有鲁棒性。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，这对于建模重尾甚至超重尾是有用的。程序的第二部分讨论了重尾分布下贝叶斯推理的鲁棒性.为了构造鲁棒性推理，建立了先验和似然的尾部条件，这意味着后验密度不受冲突信息或离群值的破坏.鲁棒性条件主要涉及先验和似然的尾部行为，特别是它表明，重尾分布是必要的。该研究项目可以在统计学、精算学、金融、保险和银行业中找到许多应用。例如，金融和精算学中存在的极端事件，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。