Bernoulli and tail-dependence compatibility

Embrechts, Paul; Hofert, Marius; Wang, Ruodu

doi:10.1214/15-aap1128

Cited by 36 publications

(37 citation statements)

References 16 publications

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“…Proof. For Θ n this property is evident from Theorem 5 and (8). But then the affine map ψ n maps Θ n to the convex polytope TCF n = ψ n (Θ n ).…”

Section: Tcf N Is a Convex Polytopementioning

confidence: 86%

“…To conclude with, independently of our research [12] and motivated from an insurance context, [8] dealt with almost the same questions (in particular Questions A and B) for random vectors with an emphasis on the construction of realizing copulas as we learned on the EVA 2015 in AnnArbor. Our approach offers (at least theoretically) an algorithm that can solve Questions A and B for random vectors completely (even though the feasibilty of such an algorithm breaks down very quickly as the dimension grows and we have doubts on its practical use in higher dimensions).…”

Section: Discussionmentioning

confidence: 99%

“…Recent developments indicate that such tools may appear more frequently in the analysis of extremes, cf. [30], [51], [49], [8] and [47].…”

Section: Introductionmentioning

confidence: 99%

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The realization problem for tail correlation functions

2016

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For a stochastic process {X t } t∈T with identical one-dimensional margins and upper endpoint τ up its tail correlation function (TCF) is defined through χ (X) (s, t) = lim τ →τup P (X s > τ | X t > τ ). It is a popular bivariate summary measure that has been frequently used in the literature in order to assess tail dependence. In this article, we study its realization problem. We show that the set of all TCFs on T × T coincides with the set of TCFs stemming from a subclass of max-stable processes and can be completely characterized by a system of affine inequalities. Basic closure properties of the set of TCFs and regularity implications of the continuity of χ are derived. If T is finite, the set of TCFs on T × T forms a convex polytope of |T | × |T | matrices. Several general results reveal its complex geometric structure. Up to |T | = 6 a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as |T | ≥ 3 grows.

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“…Proof. For Θ n this property is evident from Theorem 5 and (8). But then the affine map ψ n maps Θ n to the convex polytope TCF n = ψ n (Θ n ).…”

Section: Tcf N Is a Convex Polytopementioning

confidence: 86%

Section: Discussionmentioning

confidence: 99%

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The realization problem for tail correlation functions

2016

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“…The compatibility problems of matrices are also studied for other bivariate measures of association; see, for instance, Chaganty & Joe (2006) and Embrechts et al (2016) for compatibility of Bernoulli correlation and tail-dependence matrices.…”

Section: Known Results and Related Literaturementioning

confidence: 99%

Compatible matrices of Spearman’s rank correlation

Wang

2019

Statistics & Probability Letters

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In this paper, we provide a negative answer to a long-standing open problem on the compatibility of Spearman's rho matrices. Following an equivalence of Spearman's rho matrices and linear correlation matrices for dimensions up to 9 in the literature, we show non-equivalence for dimensions 12 or higher. In particular, we connect this problem with the existence of a random vector under some linear projection restrictions in two characterization results.

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“…In higher dimensions practitioners have begun working with matrices Λ = [λ ij ] of pairwise upper tail-dependence coefficients; see Embrechts et al (2016) for a characterization and practical application. Estimates of the entries of Λ will be taken to be the pairwise estimates λ ij .…”

Section: Tail Dependence Based On Pairwise Fitted T Copulasmentioning

confidence: 99%

Visualizing dependence in high-dimensional data: An application to S&P 500 constituent data

Hofert

Oldford

2018

Econometrics and Statistics

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The notion of a zenpath and a zenplot is introduced to search and detect dependence in high-dimensional data for model building and statistical inference. By using any measure of dependence between two random variables (such as correlation, Spearman's rho, Kendall's tau, tail dependence etc.), a zenpath can construct paths through pairs of variables in different ways, which can then be laid out and displayed by a zenplot. The approach is illustrated by investigating tail dependence and model fit in constituent data of the S&P 500 during the financial crisis of 2007-2008. The corresponding Global Industry Classification Standard (GICS) sector information is also addressed.Zenpaths and zenplots are useful tools for exploring dependence in high-dimensional data, for example, from the realm of finance, insurance and quantitative risk management. All presented algorithms are implemented using the R package zenplots and all examples and graphics in the paper can be reproduced using the accompanying demo SP500.

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Bernoulli and tail-dependence compatibility

Cited by 36 publications

References 16 publications

The realization problem for tail correlation functions

The realization problem for tail correlation functions

Compatible matrices of Spearman’s rank correlation

Visualizing dependence in high-dimensional data: An application to S&P 500 constituent data

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