A note on the Galambos copula and its associated Bernstein function

Mai, Jan-Frederik

doi:10.2478/demo-2014-0002

Cited by 3 publications

(5 citation statements)

References 15 publications

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“…4 this is proved by using socalled Eulerian numbers, counting permutations with a certain number of ascents. Another proof has been given in [6]…”

Section: Proposition Monotone Composition Theorem ("Mct")mentioning

confidence: 99%

Copulas, stable tail dependence functions, and multivariate monotonicity

Ressel

2019

Dependence Modeling

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For functions of several variables there exist many notions of monotonicity, three of them being characteristic for resp. distribution, survival and co-survival functions. In each case the “degree” of monotonicity is just the basic one of a whole scale.Copulas are special distribution functions, and stable tail dependence functions are special co-survival functions. It will turn out that for both classes the basic degree of monotonicity is the only one possible, apart from the (trivial) independence functions. As a consequence a “nesting” of such functions depends on particular circumstances. For nested Archimedean copulas the rather restrictive conditions known so far are considerably weakened.

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“…4 this is proved by using socalled Eulerian numbers, counting permutations with a certain number of ascents. Another proof has been given in [6]…”

Section: Proposition Monotone Composition Theorem ("Mct")mentioning

confidence: 99%

Copulas, stable tail dependence functions, and multivariate monotonicity

Ressel

2019

Dependence Modeling

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“…dx, and has been investigated in [21]. The associated distribution function given, for all t ∈ [0, ∞), by…”

Section: Example 2 (The Galambos Copula)mentioning

confidence: 99%

“…This parametric family is parameterized by θ ∈ (0, ∞). The Lévy measure is ν(dx) ≡ e −x /(1 − e −x ) {− ln(1 − e −x )} θ−1 /Γ(θ) dx, and has been investigated in [21]. The associated distribution function given, for all t ∈ [0, ∞), by…”

Section: Example 2 (The Galambos Copula)mentioning

confidence: 99%

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

Mai¹

2018

Journal of Multivariate Analysis

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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.

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“…, X d ) is a random vector with survival copula the Galambos copula in concern and all one-dimensional margins unit exponentially distributed. It follows from a computation in [Mai (2014)] that…”

Section: Proofmentioning

confidence: 99%

“…[Genest et al (2018), Example 5]. Further background on the Galambos copula can be found in [Mai (2014)].…”

Section: Introductionmentioning

confidence: 99%

Exact simulation of reciprocal Archimedean copulas

Mai¹

2018

Statistics & Probability Letters

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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.

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A note on the Galambos copula and its associated Bernstein function

Cited by 3 publications

References 15 publications

Copulas, stable tail dependence functions, and multivariate monotonicity

Copulas, stable tail dependence functions, and multivariate monotonicity

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

Exact simulation of reciprocal Archimedean copulas

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