Gaussian mixture copulas for flexible dependence modelling in the body and tails of joint distributions

Abstract

Fully describing the entire data set is essential in multivariate risk assessment, since moderate levels of one variable can influence another, potentially leading it to be extreme. Additionally, modelling both non-extreme and extreme events within a single framework avoids the need to select a threshold vector used to determine an extremal region, or the requirement to add flexibility to bridge between separate models for the body and tail regions. We propose a copula model, based on a mixture of Gaussian distributions, as this model avoids the need to define an extremal region, it is scalable to dimensions beyond the bivariate case, and it can handle both asymptotic dependent and asymptotic independent extremal dependence structures. We apply the proposed model through simulations and to a 5-dimensional seasonal air pollution data set, previously analysed in the multivariate extremes literature. Through pairwise, trivariate and 5-dimensional analyses, we show the flexibility of the Gaussian mixture copula in capturing different joint distributional behaviours and its ability to identify potential graphical structure features, both of which can vary across the body and tail regions.