Grouped normal variance mixtures are a class of multivariate distributions that generalize classical normal variance mixtures such as the multivariate t distribution, by allowing different groups to have different (comonotone) mixing distributions. This allows one to better model risk factors where components within a group are of similar type, but where different groups have components of quite different type. This paper provides an encompassing body of algorithms to address the computational challenges when working with this class of distributions. In particular, the distribution function and copula are estimated efficiently using randomized quasi-Monte Carlo (RQMC) algorithms. We propose to estimate the log-density function, which is in general not available in closed form, using an adaptive RQMC scheme. This, in turn, gives rise to a likelihood-based fitting procedure to jointly estimate the parameters of a grouped normal mixture copula jointly. We also provide mathematical expressions and methods to compute Kendall’s tau, Spearman’s rho and the tail dependence coefficient $\lambda$. All algorithms presented are available in the R package nvmix (version ≥ 0.0.5).

The present paper addresses computational and numerical challenges when working with t copulas and their more complicated extensions, the grouped t and skew t copulas. We demonstrate how the R package nvmix can be used to work with these copulas. In particular, we discuss (quasi-)random sampling and fitting. We highlight the difficulties arising from using more complicated models, such as the lack of availability of a joint density function or the lack of an analytical form of the marginal quantile functions, and give possible solutions along with future research ideas.

Normal variance mixtures are a class of multivariate distributions that generalize the multivariate normal by randomizing (or mixing) the covariance matrix via multiplication by a non-negative random variable W. The multivariate t distribution is an example of such mixture, where W has an inverse-gamma distribution. Algorithms to compute the joint distribution function and perform parameter estimation for the multivariate normal and t (with integer degrees of freedom) can be found in the literature and are implemented in, e.g., the R package mvtnorm. In this paper, efficient algorithms to perform these tasks in the general case of a normal variance mixture are proposed. In addition to the above two tasks, the evaluation of the joint (logarithmic) density function of a general normal variance mixture is tackled as well, as it is needed for parameter estimation and does not always exist in closed form in this more general setup. For the evaluation of the joint distribution function, the proposed algorithms apply randomized quasi-Monte Carlo (RQMC) methods in a way that improves upon existing methods proposed for the multivariate normal and t distributions. An adaptive RQMC algorithm that similarly exploits the superior convergence properties of RQMC methods is presented for the task of evaluating the joint log-density function. This allows the parameter estimation task to be accomplished via an EM-like algorithm where all weights and log-densities are numerically estimated. It is demonstrated through numerical examples that the suggested algorithms are quite fast; even for high dimensions around 1000 the distribution function can be estimated with moderate accuracy using only a few seconds of run time. Even log-densities around -100 can be estimated accurately and quickly. An implementation of all algorithms presented in this work is available in the R package nvmix (version >= 0.0.4).