Dependence modeling via copulas: inference, theoretical study and applications

Copula理论提供了强大而灵活的工具，有助于对隐含许多随机变量的现象进行建模。拟议的研究项目涉及在不同的应用环境中使用系词。它分为四个部分：第一，将开发统计方法，以便能够选择适当的Copula家族。将讨论独立和同分布观测的经典情况以及时间序列背景。所提出的技术的渐近有效性将使用经验过程的基本理论来证明。将特别注意在统计水文学领域的应用，在这一领域中，经常收集多变量与时间相关的观测数据。在第二部分中，将从随机变量和/或向量之间的渐近有效性的角度提出并评估它们之间的独立性检验。经验Copula过程的适当版本将能够在这些背景下产生有趣的独立性测试类别。在项目的第三部分，将研究多变量条件分布引起的依赖结构。我们将详细研究阿基米德分布和椭圆分布的特例。当应用条件时，这些定律表现出保持在同一类模型中的有趣性质。最后，项目的第四部分将致力于金融和精算学中的一个重要问题，即对多种资产组合的风险评估。由于缺乏关于资产之间相关性性质的信息，将为各种投资组合计算余额的风险界限。

Copula theory provides powerful and flexible tools helping to model phenomena that imply many random variables. The proposed research project concerns the use of copulas in various contexts of application. It is divided in four parts: First, statistical methods that allow the selection of an appropriate family of copulas will be developed. The classical situation of independent and identically distributed observations, as well as the time series context, will be treated. The asymptotic validity of the proposed techniques will be demonstrated using the  theory of empirical processes. Special attention will be given to applications in the domain of statistical hydrology, where multivariate time-dependent observations are often collected. In the second part, tests for the independence between random variables and/or vectors will be proposed and evaluated from the point of view of their asymptotic efficiency. Appropriate versions of the empirical copula process will enable to generate interesting classes of tests of independence in these contexts. In the third part of the project, the dependence structures induce by multivariate conditional distributions will be investigated. The special cases of Archimedean and elliptical distributions will be studied in details. These laws exhibit the interesting property of remaining in the same class of models when a conditioning is applied. Finally, the fourth part of the project will be devoted to an important problem in finance and actuarial science, namely the risk evaluation of a portfolio of many assets. Bounds on the risk that balance for the lack of information on the nature of the dependence among the assets will be computed for various portfolios.