Abstract
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 λ. All algorithms presented are available in the R package nvmix (version ≥ 0.0.5).
Highlights
It is well known that for the purpose of modeling dependence in a risk management setting, the multivariate normal distribution is not flexible enough, and its use can lead to a misleading assessment of risk(s)
The adaptive mechanism we propose ensures the estimation procedure is precise even for points that are far from the mean; (iii) to estimate Kendall’s tau and Spearman’s rho for a two-dimensional grouped NVM copula, we provide a representation as an expectation, which in turn leads to an easy-to-approximate two- or three-dimensional integral; (iv) we provide an algorithm to estimate the copula and its density associated with the grouped t copula, and fitting algorithms to estimate the parameters of a grouped NVM copula based on a dataset
The second part of this section describes randomized quasi-Monte Carlo methods, which is the type of integration method we apply to approximate quantities that are expressed as expectations, such as the distribution function of grouped normal variance mixtures
Summary
It is well known that for the purpose of modeling dependence in a risk management setting, the multivariate normal distribution is not flexible enough, and its use can lead to a misleading assessment of risk(s). Our paper fills this gap by providing a complete set of algorithms for performing the main computational tasks associated with these distributions and their associated copulas, and does so in an as automated way as possible This is done for grouped t copulas, but (in many cases) for the more general grouped normal variance mixture distributions/copulas, which allow for even further flexibility in modeling the shock variables W. The second part of this section describes randomized quasi-Monte Carlo methods, which is the type of integration method we apply to approximate quantities that are expressed as expectations, such as the distribution function of grouped normal variance mixtures
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