Title: “Copulas: A Powerful Tool to Investigate Dependencies Between Random Variables”
Abstract: “Statisticians are often required to investigate relationships between different components of random vectors of interest. The use of correlation or of analogous indexes hides important features of the dependencies between variables. The joint distribution of the random variables fully captures these dependencies, but merges them with their marginal behaviours. Copulas allow us to separate the joint and the marginal behaviour of the random variables. Sklar’s Theorem plays an important role in this framework: it relates the joint distribution to both a copula and the marginal distributions in such a way that the copula catch all the dependency features of the model. Families of copulas have been studied in the literature, where they have been extensively used for statistical model selection in the context of finance. Furthermore, many studies
have considered the stochastic ordering of different copula families. Here we illustrate some further applications of copulas, including their relationship to
mutual information estimation and their use to relate different diffusion processes. We also discuss their application for modelling purposes and a possible statistical application to detect dependencies between point processes.