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

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  • Mai, Jan-Frederik

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 ν on (0,∞]. 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.

Suggested Citation

  • Mai, Jan-Frederik, 2018. "Extreme-value copulas associated with the expected scaled maximum of independent random variables," Journal of Multivariate Analysis, Elsevier, vol. 166(C), pages 50-61.
    • Handle: RePEc:eee:jmvana:v:166:y:2018:i:c:p:50-61
    DOI: 10.1016/j.jmva.2018.02.005
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    References listed on IDEAS

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    1. Segers, Johan, 2012. "Max-stable models for multivariate extremes," LIDAM Reprints ISBA 2012012, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
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    5. Clément Dombry & Sebastian Engelke & Marco Oesting, 2016. "Exact simulation of max-stable processes," Biometrika, Biometrika Trust, vol. 103(2), pages 303-317.
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    7. Belzile, Léo R. & Nešlehová, Johanna G., 2017. "Extremal attractors of Liouville copulas," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 68-92.
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    Cited by:

    1. Mai Jan-Frederik, 2022. "About the exact simulation of bivariate (reciprocal) Archimax copulas," Dependence Modeling, De Gruyter, vol. 10(1), pages 29-47, January.
    2. Stoev, Stilian & Wang, Yizao, 2019. "Exchangeable random partitions from max-infinitely-divisible distributions," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 50-56.
    3. Mai Jan-Frederik, 2024. "Sharp bounds on the survival function of exchangeable min-stable multivariate exponential sequences," Dependence Modeling, De Gruyter, vol. 12(1), pages 1-12.

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