Semi-Parametric Probability-Weighted Moments Estimation Revisited

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 pro...

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Bibliographic Details
Published in:Methodology and computing in applied probability 2014-03, Vol.16 (1), p.1-29
Main Authors: Caeiro, Frederico, Gomes, M. Ivette, Vandewalle, Björn
Format: Article
Language:English
Subjects:
Analysis
Asymptotic properties
Business and Management
Computer science
Computer simulation
Economics
Electrical Engineering
Estimating techniques
Estimators
Extreme values
Life Sciences
Mathematical models
Mathematics and Statistics
Monte Carlo methods
Probability
Samples
Statistical methods
Statistics
Studies
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Summary: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.
ISSN:1387-5841
1573-7713
DOI:10.1007/s11009-012-9295-6