Non-Parametric Inference for Bivariate Extreme-Value Copulas

Segers, Johan

doi:10.2139/ssrn.616611

Cited by 15 publications

(28 citation statements)

References 37 publications

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“…In case of known margins, variance-minimizing weight functions λ j can be determined adaptively by ordinary least squares (Segers, 2007;Gudendorf and Segers, 2011). However, if the marginal distributions are unknown, these endpoint cor-rections are asymptotically irrelevant (Genest and Segers, 2009), since…”

Section: Nonparametric Estimation Of the Dependence Functionmentioning

confidence: 99%

Nonparametric estimation of multivariate extreme-value copulas

Gudendorf

Segers

2012

Journal of Statistical Planning and Inference

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Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estimator by its best least-squares approximation in the set of Pickands dependence functions having a discrete spectral measure supported on a sufficiently fine grid. Weak convergence of the standardized estimators is demonstrated and the finite-sample performance of the estimators is investigated by means of a simulation experiment.

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Section: Nonparametric Estimation Of the Dependence Functionmentioning

confidence: 99%

Nonparametric estimation of multivariate extreme-value copulas

Gudendorf

Segers

2012

Journal of Statistical Planning and Inference

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“…Typically, the convergence above holds in the stronger topology of uniform convergence. The limit process G is a centered Gaussian process, and ε n is equal to the reciprocal of the square root of the effective sample size, i.e., the number of block maxima (Deheuvels 1991;Capéraà, Fougères & Genest 1997;Segers 2007) or the number of high-threshold exceedances (Capéraà & Fougères 2000;Einmahl, de Haan & Piterbarg 2001). By Theorem 1, equation (14) implies that the asymptotic distribution of the projection estimatorÂ…”

Section: Moreover If the Integer Sequence M N Is Such Thatmentioning

confidence: 99%

Projection estimators of Pickands dependence functions

2008

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Abstract:The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme-value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite-dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite-dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one. Estimation par projection de la fonction de dépendance de PickandsRésumé : Les auteurs s'intéressentà la construction d'estimateurs intrinsèques de la fonction de dépendan-ce de Pickands d'une copule des valeurs extrêmes. Ils montrent comment un estimateur initial quelconque peutêtre modifié pour satisfaire les contraintes de forme voulues. Leur solution consisteà projeter cet estimateur dans l'espace des fonctions de Pickands, qui forme un sous-ensemble convexe fermé d'un espace de Hilbert. Comme la solution n'est pas explicite, ils remplacent cet espace paramétrique fonctionnel par une succession d'approximations de dimension finie. Ilsétablissent la distribution asymptotique de la projection de l'estimateur et de ses approximations de dimension finie, ce qui leur permet de conclure que l'estimateur projeté est au moins aussi efficace que l'estimateur initial.

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“…As shown in [82], the weights (1 − t) and t in de Deheuvels estimator (17) can be understood as pragmatic choices that could be replaced by suitable weight functions β 1 (t) and β 2 (t):…”

Section: Nonparametric Estimationmentioning

confidence: 99%

“…Solving the resulting differential equation for A and replacing unknown quantities by their sample versions yields the CFG-estimator. In [82] however, it was shown that the estimator admits the simpler representation…”

Section: Nonparametric Estimationmentioning

confidence: 99%

Extreme-Value Copulas

Gudendorf

Segers

2010

Lecture Notes in Statistics

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174

138

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Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in the domain of extreme-value theory, they can also be a convenient choice to model general positive dependence structures. The aim of this survey is to present the reader with the state-of-the-art in dependence modeling via extreme-value copulas. Both probabilistic and statistical issues are reviewed, in a nonparametric as well as a parametric context.

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Non-Parametric Inference for Bivariate Extreme-Value Copulas

Cited by 15 publications

References 37 publications

Nonparametric estimation of multivariate extreme-value copulas

Nonparametric estimation of multivariate extreme-value copulas

Projection estimators of Pickands dependence functions

Extreme-Value Copulas

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