Empirical Tail Copulas for Functional Data

John Einmahl, Johan Segers

Research output: Working paperDiscussion paperOther research output

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Abstract

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.
Original languageEnglish
Place of PublicationTilburg
PublisherCentER, Center for Economic Research
Number of pages32
Volume2020-004
Publication statusPublished - 3 Feb 2020

Publication series

NameCentER Discussion Paper
Volume2020-004

Fingerprint

Functional Data
Copula
Tail
Asymptotic Normality
Tail Dependence
Uniform Asymptotics
Dependence Function
Estimator
Weak Limit
Stable Distribution
Empirical Process
Domain of Attraction
Multivariate Distribution
Estimation Error
Gaussian Process
Converge
Subset
Coefficient

Keywords

  • extreme value statistics
  • functional data
  • tail empirical process
  • tal dependence
  • tial copula estimation
  • uniform asymptotic normality

Cite this

Einmahl, J., & Segers, J. (2020). Empirical Tail Copulas for Functional Data. (CentER Discussion Paper; Vol. 2020-004). Tilburg: CentER, Center for Economic Research.
Einmahl, John ; Segers, Johan. / Empirical Tail Copulas for Functional Data. Tilburg : CentER, Center for Economic Research, 2020. (CentER Discussion Paper).
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Einmahl, J & Segers, J 2020 'Empirical Tail Copulas for Functional Data' CentER Discussion Paper, vol. 2020-004, CentER, Center for Economic Research, Tilburg.

Empirical Tail Copulas for Functional Data. / Einmahl, John; Segers, Johan.

Tilburg : CentER, Center for Economic Research, 2020. (CentER Discussion Paper; Vol. 2020-004).

Research output: Working paperDiscussion paperOther research output

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AB - 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.

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Einmahl J, Segers J. Empirical Tail Copulas for Functional Data. Tilburg: CentER, Center for Economic Research. 2020 Feb 3. (CentER Discussion Paper).