A general approach to generate random variates for multivariate copulae

• Published in: Australian & New Zealand journal of statistics Vol. 60; no. 1; pp. 140 - 155
• Main Authors: Tajvidi, N., Turlach, B.A.
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.03.2018
• Subjects: Approximation; Computer simulation; Continuity (mathematics); copula; Distribution functions; extreme value distributions; Extreme values; Hastings algorithm; Mathematical analysis; Metropolis; random variate generation; Rectangles; Statistical analysis
• ISSN: 1369-1473; 1467-842X; 1467-842X
• DOI: 10.1111/anzs.12209

Summary We suggest two methods for simulating from a multivariate copula in an arbitrary dimension. Although our main emphasis in this paper is on multivariate extreme value distributions, the proposed methods can be applied to any copula. The basic idea is to approximate the (unknown) density of the copula by a distribution that has a piece‐wise constant (histogram) density. This is achieved by partitioning the support of a given copula C into a large number of hyper‐rectangles and using them to generate random variates from an approximation of the copula. We suggest two methods for finding this approximation which correspond to either finding hyper‐rectangles which have equal probability mass with respect to C, or determining a partition using hyper‐squares of equal volume and finding the corresponding probability mass of each hyper‐square. We also discuss how the generated random variates can be used as proposals in a Metropolis–Hastings algorithm, when C is an absolutely continuous distribution function, to generate a sequence of random variates from C. An implementation of the proposed methodologies is provided for the statistical computing and graphics environment R in our package called SimCop. A general approach to generating random variates from multivariate copulas which are not available in closed form.