Actuarial models and their Bayesian analysis

2013年夏天，加拿大发生了许多引人注目的悲剧和灾难，包括：阿尔伯塔洪水和卡尔加里市中心大部分地区的洪水;梅甘蒂克湖的油罐车爆炸;多伦多部分地区和周边地区的百年一遇风暴和洪水。尽管这些事件是悲惨的，但如果不是精算科学家设计的保险模型和保险政策提供的保护，它们对加拿大大多数幸存的受影响个人和企业的影响会更糟。 我的研究计划的主旨是开发新的贝叶斯统计模型和/或方法，用于感兴趣的精算科学家和精算从业人员在行业的背景下;特别是在非人寿/财产和意外伤害保险设置。贝叶斯统计方法将模型中出现的所有未知数视为随机量，并在已知信息的情况下推导出它们的分布。贝叶斯方法在精算科学中有着悠久的成功应用历史，并具有许多理想的性质。例如，它们允许将过去的经验或先前的证据纳入模型，并产生易于理解的结果。 非人寿保险索赔通常是一些非常小的，许多中型和一些非常大的价值的混合。标准分布（例如，对数正态、指数、帕累托、威布尔）通常对这样的索赔值集合提供较差的总体拟合。我的研究计划的一个目标是继续我的工作，为这些设置开发更合适的分布模型，探索这些模型的修改版本（例如，适用于已知保险索赔有上限时使用的截断版本），探索它们对多变量设置的各种扩展，并探索使用贝叶斯方法实现这些模型。改进的个人索赔额模型将使精算科学家和从业人员能够更有效地为个人保单定价。 一个保险公司通常会管理一个或多个保险线，包含数千甚至数百万的政策。对于一个给定的线，保险公司必须预测总金额（称为索赔或损失准备金）代表的钱，应该由保险公司持有，以便能够支付所有未来的索赔所产生的政策，目前生效和政策写在过去。一个特殊的情况是，当一个保险项目由几个相关的子投资组合组成时，这使得确定总损失准备金的问题变得非常复杂。我的研究计划的另一个目标是开发更好的损失准备金预测模型。 适当和公平的保险费和保险公司的健全资本化应该是我们社会大多数成员关心的问题。我的研究计划涉及这些主题。它还将开发新的统计模型和方法，这些模型和方法可能会在精算学，风险管理，生存分析和可靠性工程等各个领域中得到应用。

The summer of 2013 marked Canada with a number of high profile tragedies and disasters including: the Alberta floods, and the flooding of significant parts of downtown Calgary; the tanker train explosion in Lac Megantic; the 100-year storm and flooding in parts of Toronto and surrounding areas. As tragic as these incidents were, their impact on the majority of surviving affected individuals and businesses in Canada would have been even worse were it not for the protection afforded by insurance models and insurance policies designed by actuarial scientists. The main thrust of my research program is to develop new Bayesian statistical models and / or methods for use in contexts of interest to actuarial scientists and actuarial practitioners in industry; in particular, in non-life / property and casualty insurance settings. A Bayesian statistical method treats all unknowns appearing in a model as random quantities and derives their distribution given the known information. Bayesian methods have a long history of successful application in actuarial science and have many desirable properties. For example, they allow past experience or prior evidence to be incorporated into a model and yield results that are easily understood. Non-life insurance claims are often a mix of some very small, many mid-sized, and some extremely large values. Standard distributions (e.g., lognormal, exponential, Pareto, Weibull) often provide a poor overall fit to such a collection of claim values. One objective of my research program is to continue my work on developing more appropriate distributional models for these settings, explore modified versions of these models (e.g., truncated versions appropriate for use when insurance claims are known to be capped), explore various extensions of them to multivariate settings, and explore implementations of these models using Bayesian methods. Improved models for individual claim sizes will enable actuarial scientists and practitioners to more effectively price individual insurance policies. An insurer will often manage one or more lines of insurance containing many thousands or even millions of policies. For a given line, an insurer must forecast the aggregate amount (called the claim or loss reserve) representing the money that should be held by the insurer in order to be able to pay all future claims arising from policies currently in force and policies written in the past. A special case is when a when a line of insurance consists of several correlated subportfolios, and this enormously complicates the matter of determining the overall loss reserve. Another objective of my research program is to develop better loss reserve forecasting models. Appropriate and fair insurance premiums and the sound capitalization of insurance companies should matter and be a concern to most members of our society. My research program addresses these topics. It will also develop new statistical models and methods that are likely to find application in a variety of fields including actuarial science, risk management, survival analysis, and reliability engineering.