Sampling Archimedean copulas

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Abstract

The challenge of efficiently sampling exchangeable and nested Archimedean copulas is addressed. Specific focus is put on large dimensions, where methods involving generator derivatives are not applicable. Additionally, new conditions under which Archimedean copulas can be mixed to construct nested Archimedean copulas are presented. Moreover, for some Archimedean families, direct sampling algorithms are given. For other families, sampling algorithms based on numerical inversion of Laplace transforms are suggested. For this purpose, the Fixed Talbot, Gaver Stehfest, Gaver Wynn rho, and Laguerre series algorithm are compared in terms of precision and runtime. Examples are given, including both exchangeable and nested Archimedean copulas.

Introduction

A distinct property of Archimedean copulas is that they are fully specified by some generator function. It is important for modeling purposes that Archimedean copulas are flexible to capture various dependence structures, e.g. concordance and tail dependence. This makes them especially suitable for the modeling of extreme events. Recently, nested Archimedean copulas gained increasing interest as they extend exchangeable Archimedean copulas to allow for asymmetries, an important property e.g. in financial applications. Besides practical applications, sampling high-dimensional copulas is also interesting from a theoretical perspective.

Different methodologies for sampling bivariate Archimedean copulas are known, e.g. the conditional distribution method or an approach based on the probability integral transformation, see Embrechts et al. (2001). The former generalizes to multivariate exchangeable Archimedean copulas, see Embrechts et al. (2001), and requires the knowledge of the first d derivatives of the generator of the d-dimensional Archimedean copula under consideration. Wu et al. (2006) generalize the latter for sampling multivariate exchangeable Archimedean copulas. The resulting algorithm involves the first d1 derivatives of the Archimedean generator. A similar approach is suggested by Whelan (2004). Recently, McNeil and Nešlehová (in press) presented a sampling algorithm for multivariate exchangeable Archimedean copulas which only involves the first d2 derivatives of the generator. Still, a common drawback of all these sampling algorithms is that one has to know the generator derivatives involved. This becomes especially critical when nested Archimedean copulas are considered. The conditional distribution method is computationally impractical due to complex mixed derivatives, which is already a challenge for small dimensions, see Savu and Trede (2006) for this approach. Whelan (2004) tackles the problem of sampling nested Archimedean copulas similarly as for sampling exchangeable Archimedean copulas. His approach also requires high order derivatives, however, with respect to fewer variables than the conditional distribution method. In short, applicability of all these algorithms is strongly limited by the number of dimensions.

Considering the subclass of completely monotone Archimedean generators slightly simplifies the theory in that we have the well-known relation of generators to Laplace–Stieltjes transforms of distribution functions on the positive real line. Knowing the distribution corresponding to such a generator, Marshall and Olkin (1988) presented a sampling algorithm for exchangeable Archimedean copulas which does not require the knowledge of the copula density. This algorithm is therefore applicable to large dimensions. Algorithms for the few multivariate exchangeable Archimedean copulas that are straightforward to sample in large dimensions exploit the knowledge of the inverse Laplace–Stieltjes transform, see Joe (1997), page 375, for some examples. A generalization of the idea of Marshall and Olkin (1988) to nested Archimedean copulas and an elegant sampling algorithm was provided by McNeil (2007). However, for sampling in large dimensions, only one family is feasible, namely the Gumbel family. The reason for this is the lack of knowledge of the inverse Laplace–Stieltjes transforms of the involved generators. Further, no certain class of generators is known which can be nested such that sampling for its members is directly possible. Moreover, no example of generators belonging to different Archimedean families that can be mixed to build a nested Archimedean copula is known, which is interesting for practical applications, e.g. to allow for different kinds of tail dependence. The aim of this paper is to overcome some of these difficulties.

This paper is organized as follows. In Section 2 we present and discuss algorithms based on the inverse Laplace–Stieltjes transform for sampling exchangeable and nested Archimedean copulas. We also present techniques for the verification of the sufficient condition of McNeil (2007) for nested Archimedean structures to be proper copulas. We further obtain several new conditions which guarantee multivariate nested Archimedean copulas. For some families we obtain efficient sampling algorithms, including the cases of nested Ali-Mikhail-Haq, nested Frank, and nested Joe copulas. The section closes with examples of generators belonging to different Archimedean families that can be mixed to construct nested Archimedean copulas. In Section 3 we briefly introduce numerical inversion methods of Laplace transforms and present the Fixed Talbot, Gaver Stehfest, Gaver Wynn rho, and Laguerre series algorithm. Section 4 investigates and compares these algorithms in terms of precision and runtime using the Clayton family as reference. In Section 5 we present several examples that tackle unsolved problems. Finally, Section 6 concludes.

Section snippets

Exchangeable Archimedean copulas

An Archimedean generator is a nonincreasing, continuous function ψ:[0,][0,1] which satisfies ψ(0)=1, ψ()=0 and is strictly decreasing on [0,inf{t:ψ(t)=0}]. As McNeil and Nešlehová (in press) show, an Archimedean generator defines an exchangeable Archimedean copula, given by C(u)=C(u1,,ud;ψ)=ψ(ψ1(u1)++ψ1(ud)),u[0,1]d, if and only if ψ is d-monotone, i.e. ψ is continuous on [0,], has derivatives up to the order d2 satisfying (1)kdkdtkψ(t)0 for any k{0,,d2}, t(0,), and (1)d2dd2d

Numerical inversion of Laplace transforms

In this section, we present numerical algorithms for inverting Laplace transforms to sample exchangeable and nested Archimedean copulas. This is especially encouraged by the fact that runtime for sampling Archimedean copulas primarily depends on the number of sectors and hardly on the dimension of the copula, since uniform random numbers are easily generated. Assume as given an Archimedean generator ψΨ with ψ(t)=0etxdF(x),t[0,), and let the distribution corresponding to F be denoted by P

Comparison of the algorithms

In this section, we apply the Fixed Talbot, Gaver Stehfest, Gaver Wynn rho, and Laguerre series algorithm to evaluate F for Clayton’s family with parameter ϑ=0.8, where we have a closed-form solution as reference. This corresponds to a Kendall’s tau of 0.2857, which seems realistic for many applications. We compare the algorithms according to precision and runtime.

Examples

In this section, we present several examples of how exchangeable and nested Archimedean copulas can be sampled based on the studied techniques. For all examples, we use three, ten, and one hundred dimensions, involving only one level of nesting. However, we note that all algorithms are applicable to sample of more complex hierarchies, due to the recursive character of Algorithm 3. We further remark that for small dimensions, the conditional distribution method or other sampling algorithms may

Conclusion

In this paper, we gave sufficient conditions for the parameters of the completely monotone generators listed in Nelsen (1998), pages 94–97, such that nested Archimedean copulas can be constructed. Many of these generators fall into two categories, for which Theorem 8, Theorem 10 addressed conditions for nesting. We could generally relate the inner distribution function F0,1 involved in Theorem 8 to an exponentially tilted Stable distribution by introducing shifted Archimedean generators. As a

Acknowledgement

This paper is part of the author’s dissertation, which is carried out under the supervision of Prof. Dr. Ulrich Stadtmüller.

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