Simulating Copulas

The following sections are included:

• Dedication
• Preface
• Contents

The following sections are included:

• Copulas
  • Analytical Properties
  • Sklar’s Theorem and Survival Copulas
  • General Sampling Methodology in Low Dimensions
  • Graphical Visualization
  • Concordance Measures
  • Measures of Extremal Dependence
• General Classifications of Copulas
  • Radial Symmetry
  • Exchangeability
  • Homogeneous Mixture Models
  • Heterogeneous Mixture Models/Hierarchical Models
  • Extreme-Value Copulas

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, let 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 $F_1^{-1}$, … , $F_d^{-1}$ into the multivariate distribution function F, i.e. $C\left(u_1,\dots ,u_d\right)≔F\left(F_1^{-1}\left(u_1\right),\dots ,F_d^{-1}\left(u_d\right)\right).$ 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, for which we would like to thank them very much.

In the introductory section to this book it was noted that compared to the scarceness of work on multivariate copulas, there is extensive literature on bivariate copulas and their properties. Pair copula constructions (PCCs) build high-dimensional copulas out of bivariate ones, thus exploiting the richness of the class of bivariate copulas and providing a flexible and convenient way to extend the bivariate theory to arbitrary dimensions. This construction principle was introduced in the pioneering work of Bedford and Cooke (2001a) and Bedford and Cooke (2002), generalizing an approach by Joe (1996). Aas et al. (2009) were the first to use PCCs in an inferential context and presented a more practical approach with algorithms to calculate likelihoods and to do simulations.

This chapter was contributed by Elke Korn and Ralf Korn, 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:

• General Aspects of Generating Random Variables
• Generating Uniformly Distributed Random Variables
  • Quality Criteria for RNG
  • Common Causes of Trouble
• The Inversion Method
• Generating Exponentially Distributed Random Numbers
• Acceptance-Rejection Method
• Generating Normally Distributed Random Numbers
  • Calculating the Cumulative Normal
  • Generating Normally Distributed Random Numbers via Inversion
  • Generating Normal Random Numbers with Polar Methods
• Generating Lognormal Random Numbers
• Generating Gamma-Distributed Random Numbers
  • Generating Gamma-Distributed RNs with β < 1
  • Generating Gamma-Distributed RNs with β > 1
  • Relations to Other Distributions
• Generating Chi-Square-Distributed RNs
• Generating t-Distributed Random Numbers
• Generating Pareto-Distributed Random Numbers
• Generating Inverse Gaussian-Distributed Random Numbers
• Generating Stable-Distributed Random Numbers
• Generating Discretely Distributed Random Numbers
  • Generating Random Numbers with Geometric and Binomial Distribution
  • Generating Poisson-Distributed Random Numbers

This chapter was contributed by Elke Korn and Ralf Korn, 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:

• First Aspects of the Monte Carlo Method
• Variance Reduction Methods
  • Antithetic Variates
  • Antithetic Variates for Radially Symmetric Copulas
  • Control Variates
  • Approximation via a Simpler Dependence Structure
  • Importance Sampling
  • Importance Sampling via Increasing the Dependence
  • Further Comments on Variance Reduction Methods

One of this book’s central themes is the presentation of copula families for which the extendible subfamily can be described in convenient form. Knowledge about the stochastic model behind the extendible subfamily is of particular use for the derivation of efficient simulation algorithms, since sampling can be carried out along the associated latent one-factor model. For the families in earlier chapters the extendible subfamily has been investigated in quite some detail (e.g. Sections 2.2 and 3.3). In particular, the dependence-inducing latent factor behind the extendible Marshall–Olkin family of copulas was found to be a (possibly killed) Lévy subordinator. The present chapter presents two copula families that are both superclasses of the Marshall–Olkin family, and for which the extendible subfamilies have been investigated recently. Being generalizations of the Marshall–Olkin family of copulas, the respective extendible subfamilies are associated with stochastic processes that generalize the concept of a Lévy subordinator.

The following sections are included:

• Validating a Sampling Algorithm
• Introduction to Lévy Subordinators
  • Compound Poisson Subordinator
  • Gamma Subordinator
  • Inverse Gaussian Subordinator
  • Stable Subordinator
• Scale Mixtures of Marshall–Olkin Copulas
• Generalizations of Lévy Subordinators
  • Additive Subordinators
  • IDT Subordinators
• Further Reading

The following sections are included:

• Bibliography
• Index