Sampling

In numerous applications, copulas are widely recognized for their ability to capture complex dependencies. However, the use of copulas can pose a significant challenge due to the difficulty in accessing the related distributions and densities analytically. In such scenarios, the Monte Carlo approach provides the only viable option to obtain an approximation of the distribution of interest. Consequently, for a specific class of copulas, having a computationally efficient sampling algorithm is critical to ensure practicality and accuracy. In this chapter, we introduce a few sampling algorithms for copulas. The first algorithm, presented in Sect. 5.1, is a general approach that can be applied to any copula, but it may become computationally infeasible in high-dimensional cases. To address this limitation, we present in Sect. 5.2 a second algorithm that is specifically tailored for exchangeable ACs and is more efficient in higher dimensions. Furthermore, we demonstrate how this second algorithm can be extended to handle HACs with arbitrary structures. Such an extension is presented in Sect. 5.3, where the sampling procedure assumes complete monotonicity of all generators appearing in a HAC, as described in Sect. 2.1 . While this assumption is commonly used in applications involving HACs, as discussed in Chap. 7 , it is not strictly necessary for constructing a HAC, as outlined in Sect. 3.4 . As such, alternative sampling algorithms that do not rely on complete monotonicity are also available, and interested readers can refer to Mai (2019) for examples and a comparison with the algorithms presented below. Additionally, for readers seeking more general sampling methods for various types of copulas, we highly recommend the comprehensive review by Mai and Scherer (2012b). Finally, Sect. 5.4 presents a sampling algorithm for HACs based on outer power transformations of AC generators, discussed in Sect. 2.4 . It includes an example involving a 6-variate copula to illustrate its application.