Copulas: theory, models and methods in new directions

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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.