Multivariate risk modeling and applications

这项研究计划致力于发展随机模型和统计推断程序，以研究不同风险之间的相关性，特别是在罕见事件可能造成灾难性经济后果或影响公共安全的情况下。这一主题非常重要，例如在保险、金融或水文学中，考虑索赔、资产或极端降雨量之间的相关性对于负责任的风险管理至关重要。*分析多变量数据的经典工具，例如回归，往往依赖于风险的联合分布(或其适当的变换)是正态的不切实际的假设。一种更现实的方法包括直接通过Copula对风险之间的相关性进行建模。在处理连续数据时，Copula允许单独处理随机向量各分量之间的相关性及其边际分布。根据这一目标，并确保结论对边际分布的错误规定是稳健的，Copula的估计值和拟合优度检验通常是以等级为基础的。*赠款持有人自1985年以来一直是该地区的捐助者，并将沿着类似的路线进行他的研究。在接下来的5年里，他将构思和研究新的、灵活的、易处理的随机模型，这些模型适合分析大的相关风险集。为了确保建议的结构易于解释并很好地适应风险管理应用，他将专注于随机表示法，如随机比例和常见的冲击模型。各领域的申请将得到考虑。特别是，将开发专门适用于保险业发生的准备金和汇总索赔过程的建模技术。适用于这些模型和其他表现出联合尾部相关性的熟悉结构的基于等级的推理技术，最著名的是Archimax Copulas，将使用现代函数数据分析工具，如约束B-Spline平滑。此外，赠款持有者将对基于等级的推理技术进行雄心勃勃的扩展，以便能够使用Copula模型处理适当的离散、混合和其他非连续数据。经验多线性扩展Copula将在这些发展中发挥核心作用。*总体而言，这项工作将有助于改变统计学家、精算师和其他研究人员对多变量数据建模的方式。它将有助于不断增长的关于高维相关性建模的统计文献，并将在金融、保险、水文学和风险管理等领域产生可衡量的影响。所有新方法都将通过统计计算R项目提供给从业人员。********

This research program is concerned with the development of stochastic models and statistical inference procedures for the study of dependence between heterogeneous risks, particularly in situations where rare events could have disastrous economic consequences or affect public safety. This topic is of great importance, e.g., in insurance, finance or hydrology, where accounting for dependence between claims, assets or extreme precipitations is essential for responsible risk management.****Classical tools for analyzing multivariate data, e.g., regression, often rely on the unrealistic assumption that the joint distribution of the risks (or suitable transformations thereof) is normal. A more realistic approach consists of modeling the dependence between the risks directly through a copula. When dealing with continuous data, copulas allow for a separate treatment of the dependence between, and the marginal distributions of, the components of a random vector. In accordance with this goal, and to ensure that conclusions are robust to misspecification of the marginal distributions, estimators and goodness-of-fit tests for copulas are typically based on ranks.****The grant holder has been a contributor to the area since 1985 and will pursue his research along similar lines. Over the next 5 years, he will conceive and investigate new, flexible, and tractable stochastic models that are suitable for the analysis of large sets of dependent risks. To ensure that the proposed structures are easily interpretable and well adapted to risk management applications, he will focus on stochastic representations such as random scaling and common shock models. Applications to various fields will be considered. In particular, modeling techniques specifically adapted to reserve and aggregate claim processes occurring in the insurance industry will be developed. Rank-based inference techniques adapted to these models and other familiar structures exhibiting joint tail dependence, most notably Archimax copulas, will be designed using modern functional data analytic tools such as constrained B-spline smoothing. In addition, the grant holder will undertake an ambitious extension of rank-based inference techniques in order to be able to handle properly discrete, mixed, and otherwise non-continuous data using copula models. The empirical multi-linear extension copula will play a central role in these developments.****Overall, this work will help to change the manner in which statisticians, actuaries and other researchers model multivariate data. It will contribute to the growing statistical literature on high-dimensional dependence modeling and will have measurable impact in fields such as finance, insurance, hydrology, and risk management. All new methodology will be made accessible to practitioners through the R Project for Statistical Computing. ********