Copulas: theory, models and methods in new directions

理解随机依赖在许多科学领域都是一个非常重要的问题，包括水文学、气候学、金融学、精算科学和林业，仅举几例。在数理统计的术语中，人们想要描述一个随机向量的组成部分之间存在的联系。传统上，这类调查是基于多元正态分布和相关分布；然后通常通过Pearson相关性来量化依赖性。然而，只要随机变量之间的相关性是非线性的，这种方法就不能得出可靠的结论，这是众所周知的。处理随机依赖的现代方法依赖于copula理论，它在过去15年左右彻底改变了依赖建模的方式。这种方法的出发点是著名的Sklar定理，它基本上允许(i)通过标准方法和独立的copula建模结合边缘分布的建模来构建联合分布，以及（ii）通过只考虑其底层的copula来解决与随机向量中的依赖性相关的所有问题。

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.