Identifying groups of variables with the potential of being large simultaneously

Chiapino, Maël; Sabourin, Anne; Segers, Johan

doi:10.1007/s10687-018-0339-3

Cited by 23 publications

(15 citation statements)

References 27 publications

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“…of the stdf and all theR n,t (Chiapino, Sabourin and Segers, 2019). Hence we obtain immediately the result corresponding to Corollary 2 (in particular the weak convergence) for the empirical stdf process…”

Section: Rank-based Empirical Tail Copulassupporting

confidence: 77%

Empirical Tail Copulas for Functional Data

Einmahl

Segers

2020

SSRN Journal

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For multivariate distributions in the domain of attraction of a max-stable distribution, the tail copula and the stable tail dependence function are equivalent ways to capture the dependence in the upper tail. The empirical versions of these functions are rank-based estimators whose inflated estimation errors are known to converge weakly to a Gaussian process that is similar in structure to the weak limit of the empirical copula process. We extend this multivariate result to continuous functional data by establishing the asymptotic normality of the estimators of the tail copula, uniformly over all finite subsets of at most D points (D fixed). As a special case we obtain the uniform asymptotic normality of all estimated upper tail dependence coefficients. The main tool for deriving the result is the uniform asymptotic normality of all the D-variate tail empirical processes. The proof of the main result is non-standard.MSC 2010 subject classifications: Primary 62M99, 62G05, 62G20, 62G30, 62G32; secondary 60F17, 60G70.

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Section: Rank-based Empirical Tail Copulassupporting

confidence: 77%

Empirical Tail Copulas for Functional Data

Einmahl

Segers

2020

SSRN Journal

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“…A refinement of the latter framework is proposed in the yet unpublished work of [34]. [6] and [7] aim at identifying subgroups of components for which the probability of a joint excess over a large quantile is not negligible compared to that of an excess by a single component. [10] use graphical models to reduce the complexity of the extremal dependence structure.…”

Section: Dimensionality Reduction For Extreme Values a Brief Overviewmentioning

confidence: 99%

Principal component analysis for multivariate extremes

Drees

Sabourin

2021

Electron. J. Statist.

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In the probabilistic framework of multivariate regular variation, the first order behavior of heavy-tailed random vectors above large radial thresholds is ruled by a homogeneous limit measure. For a high dimensional vector, a reasonable assumption is that the support of this measure is concentrated on a lower dimensional subspace, meaning that certain linear combinations of the components are much likelier to be large than others. Identifying this subspace and thus reducing the dimension will facilitate a refined statistical analysis. In this work we apply Principal Component Analysis (PCA) to a re-scaled version of radially thresholded observations.Within the statistical learning framework of empirical risk minimization, our main focus is to analyze the squared reconstruction error for the exceedances over large radial thresholds. We prove that the empirical risk converges to the true risk, uniformly over all projection subspaces. As a consequence, the best projection subspace is shown to converge in probability to the optimal one, in terms of the Hausdorff distance between their intersections with the unit sphere. In addition, if the exceedances are rescaled to the unit ball, we obtain finite sample uniform guarantees to the reconstruction error pertaining to the estimated projection subspace. Numerical experiments illustrate the capability of the proposed framework to improve estimators of extreme value parameters.

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“…Chiapino and Sabourin (2017) propose an algorithm to group together nearby faces with extremal mass into feature clusters, by exploiting their graphical structure and a measure of extremal dependence. Finally, Chiapino et al (2019) extend this approach by instead using the coefficient of tail dependence of Ledford and Tawn (1996).…”

Section: Introductionmentioning

confidence: 99%

Determining the dependence structure of multivariate extremes

2020

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Summary In multivariate extreme value analysis, the nature of the extremal dependence between variables should be considered when selecting appropriate statistical models. Interest often lies in determining which subsets of variables can take their largest values simultaneously while the others are of smaller order. Our approach to this problem exploits hidden regular variation properties on a collection of nonstandard cones, and provides a new set of indices that reveal aspects of the extremal dependence structure not available through existing measures of dependence. We derive theoretical properties of these indices, demonstrate their utility through a series of examples, and develop methods of inference that also estimate the proportion of extremal mass associated with each cone. We apply the methods to river flows in the U.K., estimating the probabilities of different subsets of sites being large simultaneously.

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Identifying groups of variables with the potential of being large simultaneously

Cited by 23 publications

References 27 publications

Empirical Tail Copulas for Functional Data

Empirical Tail Copulas for Functional Data

Principal component analysis for multivariate extremes

Determining the dependence structure of multivariate extremes

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