Practical Notes On Multivariate Modeling Based on Elliptical Copulas

Abstract

Multivariate distributions based on elliptical copulas have been widely used in many fields such as hydrology and finance. We focus on two practical issues of applications of such models. The first is a caveat rooted in a consistency property defined by Kano (1994, Journal of Multivariate Analysis, 51:139–147) for elliptical distributions. Some elliptical families do not have this property, which puts practical limitations on applications and software implementation of the corresponding elliptical copulas. The second issue is on conditional sampling from such distributions, which is important in Monte Carlo statistical inferences, especially when closed-form solutions are not available or feasible. Two sampling methods are presented: a direct sampling approach based on a stochastic representation of elliptical distributions, and an acceptance/rejection sampling method. The latter also provides an importance sampler as a byproduct, which may have higher efficiency for some applications. A trivariate model of the volume, duration, and peak intensity of annual extreme storms illustrates the sampling algorithms.