Dependence and uncertainty: a copula-based framework

Abstract

In the era of machine learning and artificial intelligence, multivariate statistical analysis has become an inevitable tool due to the increasing complexity and dimensionality of data arising from diverse domains such as engineering, medicine, finance, and environmental science. Unlike univariate techniques, multivariate analysis provides a comprehensive framework to model, interpret, and infer the relationships among multiple random variables simultaneously. Two crucial aspects of multivariate analysis are (i) the marginal behaviour of each component and (ii) the dependence structure among the variables. One of the main challenges in multivariate statistical analysis lies in the flexible and accurate representation of this dependence structure. Classical approaches, such as the multivariate normal distribution, often rely on strict assumptions such as linearity, which seldom occurs in reality. These limitations are particularly evident in the presence of non-linear relationships, asymmetries, or tail dependencies.