Frankʼs condition for multivariate Archimedean copulas

Erdely, Arturo; González-Barrios, José M.; Hernández-Cedillo, María M.

doi:10.1016/j.fss.2013.05.017

Cited by 9 publications

(6 citation statements)

References 10 publications

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“…Remark 10. From the equivalence between signature representation and properties (i) and (iv) we can conclude that, under our conditions, the same information contained in G 1:r , ..., G r:r is also contained in the family of all the reliability functions R (ϕ) S (t) in (10). We emphasize that such a family is indexed by all the coherent structures ϕ for binary systems made of r components.…”

Section: Diagonal Sections and Distributions Of Order Statisticsmentioning

confidence: 57%

Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

Foschi¹,

Nappo²,

Spizzichino³

2021

Preprint

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As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables T 1 , ..., T r . In any case, we assume that T 1 , ..., T r are identically distributed, with a common survival function G and their survival copula is denoted by K. The diagonal's and subdiagonals' sections of K, along with G, are possible tools to describe the information needed to recover the laws of order statistics.When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of T 1 , ..., T r also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distributions of interest. This study also leads us to compare the two cases of exchangeable and minimally stable variables both in terms of copulas and of m.c.h.r. functions. The paper concludes with the analysis of two remarkable special cases of stochastic dependence, namely Archimedean copulas and load sharing models. This analysis will allow us to provide some illustrative examples, and some discussion about peculiar aspects of our results.

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Section: Diagonal Sections and Distributions Of Order Statisticsmentioning

confidence: 57%

Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

Foschi¹,

Nappo²,

Spizzichino³

2021

Preprint

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show abstract

“…We recall that the diagonal section characterizes uniquely many Archimedean copulas (under a condition that is called Frank's condition, see e.g., [17]), some non-parametric estimators of the generator of an Archimedean copulas directly rely on this diagonal section. We consider here the case where the df of Z has spectral random vector W.…”

Section: Domination Spectrummentioning

confidence: 99%

Asymptotic domination of sample maxima

Hashorva

Rullière

2020

Statistics & Probability Letters

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For a given random sample from some underlying multivariate distribution F we consider the domination of the component-wise maxima by some independent random vector W with distribution function G. We show that the probability that certain components of the sample maxima are dominated by the corresponding components of W can be approximated under the assumptions that both F and G are in the max-domain of attraction of some max-stable distribution functions. We study further some basic probabilistic properties of the dominated components of sample maxima by W .

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“…Under what is called Frank's condition (see [17]), the Archimedean copula is uniquely determined by its diagonal section δ(u) = C(u, . .…”

Section: Estimation In the Archimedean Casementioning

confidence: 99%

On the estimation of Pareto fronts from the point of view of copula theory

Binois

Rullière

Roustant

2015

Information Sciences

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International audienceGiven a first set of observations from a design of experiments sampled randomly in the design space, the corresponding set of non-dominated points usually does not give a good approximation of the Pareto front. We propose here to study this problem from the point of view of multivariate analysis, introducing a probabilistic framework with the use of copulas. This approach enables the expression of level lines in the objective space, giving an estimation of the position of the Pareto front when the level tends to zero. In particular, when it is possible to use Archimedean copulas, analytical expressions for Pareto front estimators are available. Several case studies illustrate the interest of the approach, which can be used at the beginning of the optimization when sampling randomly in the design space

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Frankʼs condition for multivariate Archimedean copulas

Cited by 9 publications

References 10 publications

Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

Asymptotic domination of sample maxima

On the estimation of Pareto fronts from the point of view of copula theory

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