Simulating Copulas

The following sections are included:

• Preface

• Contents

Before we start, let us clarify some notations…

This chapter on Archimedean copulas is organized as follows (see Figure 2.1). After a brief motivation in Section 2.1, Section 2.2 studies extendible Archimedean copulas. Those allow for a stochastic representation with components that are conditionally i.i.d. in the sense of de Finetti's theorem (see Theorem 1.4). Subsequently, Section 2.3 studies the more general family of exchangeable Archimedean copulas. This family additionally allows for negative association between components. Intuitively, this family "shrinks" in the dimension d, with the extendible class as limit for d → ∞. Finally, Section 2.4 shows how to construct and simulate h-extendible (hierarchical) Archimedean copulas. This is important in order to take into account non-exchangeable dependence patterns without losing some advantages of the (extendible) Archimedean class…

This chapter on Marshall—Olkin copulas is organized as follows (see Figure 3.1). First, Section 3.1 introduces the most general form of Marshall—Olkin copulas. Second, Section 3.2 explicitly treats the subfamily of exchangeable Marshall—Olkin copulas. These have the advantage that the functional form and the respective sampling algorithms are much simpler and thus more efficient compared to the general case. Section 3.2.4 treats hierarchical versions of Marshall—Olkin copulas, constructed from exchangeable building blocks. These lie between the most general and the exchangeable Marshall—Olkin family. Finally, Section 3.3 studies the subfamily of (exchangeable and) extendible Marshall—Olkin copulas, which are also called Lévy-frailty copulas. These are even more efficient to simulate due to their latent factor structure. Furthermore, they give rise to quite convenient parameterizations of Marshall—Olkin copulas via so-called Laplace exponents of Lévy subordinators. Moreover, it is indicated in Section 3.3.4 how h-extendible Marshall—Olkin copulas can be constructed from extendible building blocks. This technique allows us to design parametric subfamilies of Marshall—Olkin copulas that are convenient to parameterize, flexible, and efficient to simulate…

The class of elliptical copulas differs from most aforementioned classes of copulas in the way that only an implicit analytical expression is available. More precisely, an elliptical copula is defined via Sklar's theorem as the dependence structure of a related elliptical distribution and is obtained from the respective multivariate distribution function by standardizing the univariate marginal laws. Sampling such copulas therefore requires a sampling scheme for the respective multivariate distribution and an analytical expression of the univariate marginal distribution functions. To make this more precise, let¹ X = (X₁,…,X_d)′ be a sample from some elliptical distribution and let F₁,…,F_d be the univariate marginal laws. Then, (F₁(X₁),…,F_d(X_d)′ is a sample from the respective copula. The analytical form of the copula is obtained by plugging the univariate marginal quantile functions into the multivariate distribution function F, i.e. . It is only rarely possible to simplify this expression, since the univariate quantiles are often not available in closed form and the multivariate distribution function is typically complicated. The most prominent example of the class of elliptical copulas is the Gaussian copula (see Example 1.7 of Chapter 1 and Section 4.5). Another important example is the t-copula (see Section 4.5)…

This chapter was contributed by Claudia Czado and Jakob Stöber,¹ for which we would like to thank them very much.²…

This chapter was contributed by Elke Korn and Ralf Korn,¹ for which we would like to thank them very much. Much more on the subject matter is to be found in their recent monograph Korn et al. (2010)…

This chapter was contributed by Elke Korn and Ralf Korn,¹ for which we would like to thank them very much. Much more on the subject matter is to be found in their recent monograph Korn et al. (2010)…

The following sections are included:

• Appendix A Supplemental Material

• Bibliography

• Index