Semi-Parametric Probability-Weighted Moments Estimation Revisited

  • Frederico Caeiro
  • M. Ivette Gomes
  • Björn Vandewalle


In this paper, for heavy-tailed models and through the use of probability weighted moments based on the largest observations, we deal essentially with the semi-parametric estimation of the Value-at-Risk at a level p, the size of the loss occurred with a small probability p, as well as the dual problem of estimation of the probability of exceedance of a high level x. These estimation procedures depend crucially on the estimation of the extreme value index, the primary parameter in Statistics of Extremes, also done on the basis of the same weighted moments. Under regular variation conditions on the right-tail of the underlying distribution function F, we prove the consistency and asymptotic normality of the estimators under consideration in this paper, through the usual link of their asymptotic behaviour to the one of the extreme value index estimator they are based on. The performance of these estimators, for finite samples, is illustrated through Monte-Carlo simulations. An adaptive choice of thresholds is put forward. Applications to a real data set in the field of insurance as well as to simulated data are also provided.


Heavy tails Value-at-risk or high quantiles Probability of exceedance of a high level Semi-parametric estimation 

AMS 2000 Subject Classifications

Primary 62G32 62E20; Secondary 65C05 


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Frederico Caeiro
    • 1
  • M. Ivette Gomes
    • 2
  • Björn Vandewalle
    • 3
  1. 1.FCT and CMAUniversidade Nova de LisboaLisbonPortugal
  2. 2.DEIO - CEAUL and FCULUniversidade de LisboaLisbonPortugal
  3. 3.ISEGI and CEAULUniversidade Nova de LisboaLisbonPortugal

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