Unbiased Tail Estimation by an Extension of the Generalized Pareto Distribution

CentER Discussion Paper No. 2005-112

17 Pages Posted: 14 Nov 2005

See all articles by Jan Beirlant

Jan Beirlant

Catholic University of Leuven (KUL)

Elisabeth Joossens

Joint Research Centre; Catholic University of Leuven (KUL)

Johan Segers

Catholic University of Louvain (UCL)

Date Written: October 21, 2005

Abstract

The generalized Pareto distribution (GPD) is probably the most popular model for inference on the tail of a distribution. The peaks-over-threshold methodology postulates the GPD as the natural model for excesses over a high threshold. However, for the GPD to fit such excesses well, the threshold should often be rather large, thereby restricting the model to only a small upper fraction of the data. In case of heavy-tailed distributions, we propose an extension of the GPD with a single parameter, motivated by a second-order refinement of the underlying Pareto-type model. Not only can the extended model be fitted to a larger fraction of the data, but in addition is the resulting maximum likelihood for the tail index asymptotically unbiased. In practice, sample paths of the new tail index estimator as a function of the chosen threshold exhibit much larger regions of stability around the true value. We apply the method to daily log-returns of the euro-UK pound exchange rate. Some simulation results are presented as well.

Keywords: bias reduction, exchange rate, heavy tails, peaks-over-threshold, regular variation, tail index

JEL Classification: C13, C14

Suggested Citation

Beirlant, Jan and Joossens, Elisabeth and Segers, Johan, Unbiased Tail Estimation by an Extension of the Generalized Pareto Distribution (October 21, 2005). CentER Discussion Paper No. 2005-112. Available at SSRN: https://ssrn.com/abstract=845385 or http://dx.doi.org/10.2139/ssrn.845385

Jan Beirlant

Catholic University of Leuven (KUL) ( email )

W. de Croylaan 54
Leuven, B-3001
Belgium

Elisabeth Joossens

Joint Research Centre ( email )

Via E. Fermi 1
I-21020 Ispra (VA)
Italy

Catholic University of Leuven (KUL) ( email )

Oude Markt 13
Leuven, Vlaams-Brabant
Belgium

Johan Segers (Contact Author)

Catholic University of Louvain (UCL) ( email )

Place Montesquieu, 3
Louvain-la-Neuve, 1348
Belgium
+32 10 474311 (Phone)
+32 10 473032 (Fax)

HOME PAGE: http://www.uclouvain.be/stat

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