Extreme-value copulas associated with the expected scaled maximum of independent random variables

Abstract

It is well-known that the expected scaled maximum of non-negative random variables with unit mean defines a stable tail dependence function associated with some extreme-value copula. In the special case when these random variables are independent and identically distributed, min-stable multivariate exponential random vectors with the associated survival extreme-value copulas are shown to arise as finite-dimensional margins of an infinite exchangeable sequence in the sense of De Finetti's Theorem. The associated latent factor is a stochastic process which is strongly infinitely divisible with respect to time, which induces a bijection from the set of distribution functions F of non-negative random variables with finite mean to the set of Lévy measures on the positive half-axis. Since the Gumbel and the Galambos copula are the most popular examples of this construction, the investigation of this bijection contributes to a further understanding of their well-known analytical similarities. Furthermore, a simulation algorithm based on the latent factor representation is developed, if the support of F is bounded. Especially in large dimensions, this algorithm is efficient because it makes use of the De Finetti structure.

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Cited by in corpus (6)

• The infinite extendibility problem for exchangeable real-valued random vectors
• Canonical spectral representation for exchangeable max-stable sequences
• Subordinators which are infinitely divisible w.r.t. time: Construction, properties, and simulation of max-stable sequences and infinitely divisible laws
• Limit distributions of the upper order statistics for the Lévy-frailty Marshall-Olkin distribution
• Superposition, reduction of multivariable problems, and approximation
• Exchangeable random partitions from max-infinitely-divisible distributions