Partially nested Archimedean copulas in the stochastic modeling of extremal dependence

Abstract

Multivariate statistical analysis requires the construction of multidimensional dependence functions and asymptotic estimation of extremal dependence coefficients. Copulas, in particular Archimedean copulas, have attractive properties favoring enormous applications in modeling. This study aims to use Archimidian copulas to construct dependence functions including temporal dynamics. The probability that the best performance of a series of actions will be affected by an extremely poor performance of one action and vice-versa can be measured by the extremal dependence coefficients. These coefficients can be easily determined for classical Archimedean copulas, their determination for hierarchical Archimedean copulas requires a multi-step procedure. Thus, within the framework of this study, a recursive approach allowing the determination of the lower and upper extremal dependence coefficients of a hierarchical Archimedean copula model is developed. Indeed, This study proposes a multidimensional dependence function constructed using extreme value theory and nested Archimedean copulas. The notion of regular variation  is used to estimate the extremal dependence coefficients of the nested Archimedean copula model.