Tail order and intermediate tail dependence of multivariate copulas

Hua, Lei; Joe, Harry

doi:10.1016/j.jmva.2011.05.011

Cited by 119 publications

(98 citation statements)

References 26 publications

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“…While, in [7], a heuristic argument using the monotone density theorem (Theorem 1.7.2 of [18]) is used for derivatives of a tail order function, and ultimate monotonicity of a function needs to be checked. Here we give a proof of the result only on the bivariate case for tail order functions to avoid tedious arguments, and the proof is based on the typical arguments for the monotone density theorem.…”

Section: Tail Order Density Of Liouville Copulasmentioning

confidence: 99%

“…For a bivariate Gumbel copula that has the form C(u, v) = exp{−A(− log u, − log v)}, where A(x, y) = (x δ + y δ ) 1/δ , 1 ≤ δ < ∞. Then, based on Example 2 of [7],…”

Section: Tail Order Density Of Liouville Copulasmentioning

confidence: 99%

“…. , w d ) is referred to as the (lower) tail order density function of C. We refer to [7,8] for details about the notion of tail orders. The notion of tail order of copulas corresponds to 1/η, with η proposed in [9] as an "coefficient of tail dependence" when the univariate marginals follow the unit Fréchet distribution.…”

Section: Introductionmentioning

confidence: 99%

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A Note on Upper Tail Behavior of Liouville Copulas

Hua¹

2016

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Abstract:The family of Liouville copulas is defined as the survival copulas of multivariate Liouville distributions, and it covers the Archimedean copulas constructed by Williamson's d-transform. Liouville copulas provide a very wide range of dependence ranging from positive to negative dependence in the upper tails, and they can be useful in modeling tail risks. In this article, we study the upper tail behavior of Liouville copulas through their upper tail orders. Tail orders of a more general scale mixture model that covers Liouville distributions is first derived, and then tail order functions and tail order density functions of Liouville copulas are derived. Concrete examples are given after the main results.

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Section: Tail Order Density Of Liouville Copulasmentioning

confidence: 99%

“…For a bivariate Gumbel copula that has the form C(u, v) = exp{−A(− log u, − log v)}, where A(x, y) = (x δ + y δ ) 1/δ , 1 ≤ δ < ∞. Then, based on Example 2 of [7],…”

Section: Tail Order Density Of Liouville Copulasmentioning

confidence: 99%

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A Note on Upper Tail Behavior of Liouville Copulas

Hua¹

2016

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“…The reader is referred to the monographs by Joe [2] and Nelsen [3] for detailed accounts of the theory and surveys of commonly used copulas and to the review papers [4][5][6], work on tail dependence [7,8], and papers on applications [9,10]. Some work has been done for constructing asymmetric copulas ( [11,12], for example).…”

Section: Construction Of Asymmetric Copulasmentioning

confidence: 99%

Construction of asymmetric copulas and its application in two-dimensional reliability modelling

2014

European Journal of Operational Research

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Copulas offer a useful tool in modelling the dependence among random variables. In the literature, most of the existing copulas are symmetric while data collected from the real world may exhibit asymmetric nature. This necessitates developing asymmetric copulas that can model such data. In the meantime, existing methods of modelling two-dimensional reliability data are not able to capture the tail dependence that exists between the pair of age and usage, which are the two dimensions designated to describe product life. This paper proposes a new method of constructing asymmetric copulas, discusses the properties of the new copulas, and applies the method to fit two-dimensional reliability data that are collected from the real world.

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“…The Pearson-Kotz Dirichlet random vectors are another important subclass of the Dirichlet class including two prominent cases: the Student t random vectors (see e.g., [24] or [19] and the Pearson Type VII L p random vectors introduced by Gupta and Song [14]. Interestingly, the Pearson-Kotz Dirichlet random vectors are a special case of the scale mixtures of the Kotz-Dirichlet random vectors.…”

Section: Introductionmentioning

confidence: 99%

Scale mixtures of Kotz–Dirichlet distributions

Hashorva¹

2013

Journal of Multivariate Analysis

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a b s t r a c tIn this paper, we first show that a k-dimensional Dirichlet random vector has independent components if and only if it is a Kotz Type I Dirichlet random vector. We then consider in detail the class of k-dimensional scale mixtures of Kotz-Dirichlet random vectors, which is a natural extension of the class of Kotz Type I random vectors. An interesting member of the Kotz-Dirichlet class of multivariate distributions is the family of Pearson-Kotz Dirichlet distributions, for which we present a new distributional property. In an asymptotic framework, we show that the Kotz Type I Dirichlet distributions approximate the conditional distributions of scale mixtures of Kotz-Dirichlet random vectors. Furthermore, we show that the tail indices of regularly varying Dirichlet random vectors can be expressed in terms of the Kotz Type I Dirichlet random vectors.

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Tail order and intermediate tail dependence of multivariate copulas

Cited by 119 publications

References 26 publications

A Note on Upper Tail Behavior of Liouville Copulas

A Note on Upper Tail Behavior of Liouville Copulas

Construction of asymmetric copulas and its application in two-dimensional reliability modelling

Scale mixtures of Kotz–Dirichlet distributions

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