Copulas: theory, models and methods in new directions

理解随机相关性在水文学、气候学、金融学、精算学和林业等许多科学领域都是一个非常重要的问题。在数理统计的术语中，人们希望描述随机向量的分量之间存在的联系。传统上，这样的调查是基于多变量正态分布和相关分布;依赖性通常通过皮尔逊相关性来量化。然而，当随机变量之间的依赖性是非线性的，这是众所周知的，这样的方法不能产生可靠的结论。处理随机相关性的现代方法依赖于Copula理论，在过去的15年左右，Copula理论彻底改变了相关性建模的方式。这种方法的出发点是著名的Sklar定理，它基本上允许（i）通过标准方法结合边际分布的建模和独立的copula建模来构建联合分布，以及（ii）随机向量的相依性问题是指随机向量中的Copula函数，近年来在理论和实践上都取得了重大进展Copula模型的相关性度量、拟合优度检验、极值理论、参数估计和序列相关多元观测的扩展。尽管Copula方法取得了许多成功，并且对Copula感兴趣的研究人员非常活跃，但需要详细分析依赖关系的几个领域仍然几乎完全未被探索。这是这个雄心勃勃的研究计划的目的是开发有用的依赖模型和强有力的理由统计工具，在新的应用环境。该项目分为四个独立的部分，即（I）依赖结构中的变点检测，（II）条件copula模型中的推理，（III）用copula建模空间依赖和（IV）探索copula在新领域，例如经验似然和函数数据。

Understanding stochastic dependence is a very important issue in many scientific fields including hydrology, climatology, finance, actuarial sciences and forestry, to name only a few. In the terminology of mathematical statistics, one wants to describe the links that exist between the components of a random vector. Traditionally, such investigations were based on the multivariate Normal and related distributions; dependence is then typically quantified via the Pearson correlations. However, whenever the nature of the dependence between random variables is nonlinear, it is well known that such an approach fails to yield reliable conclusions. The modern way to deal with stochastic dependence relies on copula theory, which in the last fifteen years or so revolutionized the way dependence modeling is done. The starting point of this approach is the celebrated Theorem of Sklar, which basically allows to (i) build joint distributions by combining the modeling of the marginal distributions through standard methods and, independently, copula modeling, and (ii) address all questions related to dependence in a random vector by considering exclusively its underlying copula.The recent years have seen significant theoretical and practical advances on various aspects of copula modeling, including dependence measures, goodness-of-fit tests, extreme-value theory, parameter estimation and extensions to serially dependent multivariate observations. Despite the many successes of the copula approach and the hyperactivity of researchers interested in copulas, several fields that require a detailed analysis of dependence remain almost completely unexplored. It is the aim of this ambitious research program to develop useful dependence models and strongly justified statistical tools in new contexts of applications. The project is divided into four independent parts, namely (I) Change-point detection in dependence structures, (II) Inference in conditional copula models, (III) Modeling spatial dependence with copulas and (IV) Exploration of copulas in new territories, e.g. empirical likelihood and functional data.