Hierarchical Archimedean copulas through multivariate compound distributions

Cossette, Hélène; Gadoury, Simon-Pierre; Marceau, Étienne; Mtalai, Itre

doi:10.1016/j.insmatheco.2017.06.001

Cited by 11 publications

(5 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The defining properties of the Dirichlet prior imply that the assumptions of Lemma 1.7 are satisfied, which implies the claim. 16 Recall from Remark 3. family of so-called multivariate phase-type distributions, see [5]. Which members of theses families of distributions are conditionally iid?…”

Section: Example 62 (A Simple Global Shock Model)mentioning

confidence: 99%

“…For instance, nested and hierarchical extensions of (exchangeable) Archimedean copulas have become quite popular, see, e.g. [16,47,49,82,107,65].…”

Section: D}mentioning

confidence: 99%

“…To this end, let g 2 be a strictly increasing and continuous distribution function of some random variable taking values in [0, 1], assuming x → g 2 (x)/x is non-increasing on (0, 1]. The function F M (x) := x/g 2 (x) then is a distribution function on [0, 1], and we let M be a has a Dirichlet distribution 16 It is insightful to remark that for c 0 the copula Ĉc converges to the so-called upper-Fréchet Hoeffding copula Ĉ0 (u) = u [1] , and for c ∞ to the copula Ĉ∞ (u) = d k=1 u k associated with independence. The intuition of the Dirichlet prior model is that all components of X have distribution function G, but one is uncertain whether G is really the correct distribution function.…”

Section: Example 62 (A Simple Global Shock Model)mentioning

confidence: 99%

See 2 more Smart Citations

The infinite extendibility problem for exchangeable real-valued random vectors

Mai¹

2020

Probab. Surveys

View full text Add to dashboard Cite

We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on R d , which is: Can a given random vector X = (X 1 , . . . , X d ) be represented in distribution as the first d members of an infinite exchangeable sequence of random variables? This is the case if and only if X has a stochastic representation that is "conditionally iid" according to the seminal de Finetti's Theorem. Of particular interest are cases in which the original motivation behind the model X is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when X takes values in {0, 1} d , has a spherical law, a law with 1 -norm symmetric survival function, or a law with ∞-norm symmetric density. The solutions in all these cases constitute analytical characterizations of mixtures of iid sequences drawn from popular, one-parametric probability laws on R, like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when X has a Marshall-Olkin distribution, a multivariate wide-sense geometric distribution, a multivariate extreme-value distribution, or is defined as a certain exogenous shock model including the special case when its components are samples from a Dirichlet prior. The solutions in these cases correspond to iid sequences drawn from random distribution functions defined in terms of popular families of non-decreasing stochastic processes, like a Lévy subordinator, a random walk, a process that is strongly infinitely divisible with respect to time, or an additive process. The survey finishes with a list of potentially interesting open problems. In comparison to former literature on the topic, this survey purposely dispenses with generalizations to the related and larger concept of finite exchangeability or to more general state spaces than R. Instead, it aims to constitute an up-to-date comprehensive collection of known and compelling solutions of the real-valued extendibility problem, accessible for both applied and theoretical probabilists, presented in a lecture-like fashion.

show abstract

Section: Example 62 (A Simple Global Shock Model)mentioning

confidence: 99%

“…For instance, nested and hierarchical extensions of (exchangeable) Archimedean copulas have become quite popular, see, e.g. [16,47,49,82,107,65].…”

Section: D}mentioning

confidence: 99%

Section: Example 62 (A Simple Global Shock Model)mentioning

confidence: 99%

See 1 more Smart Citation

The infinite extendibility problem for exchangeable real-valued random vectors

Mai¹

2020

Probab. Surveys

View full text Add to dashboard Cite

show abstract

“…For instance, nested and hierarchical extension of (exchangeable) Archimedean copulas have become quite popular, see, e.g. [15,43,44,72,93,58].…”

Section: Remark 38 (Archimedean Copulas)mentioning

confidence: 99%

“…Example 6.6 (A one-parametric, multivariate Pareto distribution) Let Ψ(x) = α log(1 + x) be the Bernstein function associated with a Gamma distribution 15 with parameter α > 0. The Gamma distribution is self-decomposable and the survival function (42) takes the explicit, one-parametric form…”

Section: Proofmentioning

confidence: 99%

The infinite extendibility problem for exchangeable real-valued random vectors

Mai

2019

Preprint

View full text Add to dashboard Cite

We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on R d , which is: Can a given random vector X = (X 1 , . . . , X d ) be represented in distribution as the first d members of an infinite exchangeable sequence of random variables? This is the case if and only if X has a stochastic representation that is "conditionally iid" according to the seminal De Finetti's Theorem. Of particular interest are cases in which the original motivation behind the model X is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when X takes values in {0, 1} d , has a spherical law, a law with 1 -norm symmetric survival function, or a law with ∞ -norm symmetric density. The solutions in all these cases constitute analytical characterizations of mixtures of iid sequences drawn from popular, one-parametric probability laws on R, like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when X has a Marshall-Olkin distribution, a multivariate wide-sense geometric distribution, a multivariate extreme-value distribution, or is defined as a certain exogenous shock model including the special case when its components are samples from a Dirichlet prior. The solutions in these cases correspond to iid sequences drawn from random distribution functions defined in terms of popular families of non-decreasing stochastic processes, like a Lévy subordinator, a random walk, a process that is strongly infinitely divisible with respect to time, or an additive process. The survey finishes with a list of potentially interesting open problems. In comparison to former literature on the topic, this survey purposely dispenses with generalizations to the related and larger concept of finite exchangeability or to more general state spaces than R. Instead, it aims to constitute an up-to-date comprehensive collection of known and compelling solutions of the real-valued extendibility problem, accessible for both applied and theoretical probabilists, presented in a lecture-like fashion.

show abstract