Exact simulation of reciprocal Archimedean copulas

Abstract

The decreasing enumeration of the points of a Poisson random measure whose mean measure is Radon on (0, ∞] can be represented as a non-increasing function of the jump times of a standard Poisson process. This observation allows to generalize the essential idea from a well-known exact simulation algorithm for arbitrary extreme-value copulas to copulas of a more general family of max-infinitely divisible distributions, with reciprocal Archimedean copulas being a particular example.

Exact simulation of continuous max-id processes

“…Proof. By (10), µ j is a finite measure for each j ∈ J 0 . Thus, Xj is obtained by the simulation of a finite PRM with intensity µ j .…”

“…For example, a common representation of an exponent measure of a max-id random vector is a scale mixture of a probability distribution on the non-negative unit sphere of some norm on R d . Two famous representatives of this class of exponent measures are the exponent measures of max-stable random vectors with unit Fréchet margins [16, Chapter 5] and random vectors with reciprocal Archimedean copula [7,10], see Example 3.10 below. In both cases, a simulation of X via Algorithm 1 would require to deviate from the natural description of the exponent measure to simulate the PRM with intensity 1 {f (t)≥c} dµ(f ).…”

“…assume that h X = 0. Our goal is to generalize the algorithms of [20,4,10] to max-id random vectors. Similar to Algorithm 1 we will only simulate those atoms of N which are relevant to determine max x∈N x = X.…”

“…Proof. The µ j are finite intensity measures by their definition in (10). Therefore, Algorithm 2 stops after finitely many steps if and only if the while-loop from lines 7-11 stops after finitely many steps.…”

“…Setting µ 1 = s −2 ds and • = • 1 yields the family of max-stable distributions with unit Fréchet margins [16, Section 5], whereas setting • = • 1 and µ 2 to the uniform distribution on S • 1 yields the family of distributions with reciprocal Archimedean copula and marginal distribution function exp (−µ 1 ((•, ∞))) [10].…”