Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T09:54:50.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of second order parameters using probability weighted moments

Published online by Cambridge University Press:  03 July 2012

Julien Worms  and
Rym Worms
Julien Worms
Affiliation:
Universitéde Versailles Saint-Quentin, Laboratoire de Mathématiques de Versailles (CNRS UMR 8100), UFR de Sciences, Bât. Fermat, 45 Av. des Etats-Unis, 78035 Versailles Cedex, France. [email protected]
Rym Worms
Affiliation:
Université Paris Est Créteil, Laboratoire d’Analyse et de Mathématiques Appliquées (CNRS UMR 8050), 61 Av. du Gl de Gaulle, 94010 Créteil Cedex, France; [email protected]
Get access

Abstract

The P.O.T. method (Peaks Over Threshold) consists in using the generalized Pareto distribution (GPD) as an approximation for the distribution of excesses over a high threshold. In this work, we use a refinement of this approximation in order to estimate second order parameters of the model using the method of probability-weighted moments (PWM): in particular, this leads to the introduction of a new estimator for the second order parameter ρ, which will be compared to other recent estimators through some simulations. Asymptotic normality results are also proved. Our new estimator of ρ looks especially competitive when  |ρ|  is small.

Keywords

Extreme valuesdomain of attractionexcessesgeneralized Pareto distributionprobability-weighted momentssecond order parameterthird order condition.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 97 - 113
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Balkema, A. and de Haan, L., Residual life time at a great age. Ann. Probab. 2 (1974) 792–801. Google Scholar
Caeiro, F., Gomes, M.I. and Pestana, D., A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator. Stat. Probab. Lett. 79 (2009) 295–303. Google Scholar
Ciuperca, G. and Mercadier, C., Semi-parametric estimation for heavy tailed distributions. Extremes 13 (2010) 55–87. Google Scholar
Diebolt, J., Guillou, A. and Worms, R., Asymptotic behaviour of the probability-weighted moments and penultimate approximation. ESAIM : PS 7 (2003) 217–236. Google Scholar
Diebolt, J., Guillou, A. and Rached, I., Approximation of the distribution of excesses through a generalized probability-weighted moments method. J. Statist. Plann. Inference 137 (2007) 841–857. Google Scholar
Diebolt, J., Guillou, A. and Rached, I., Approximation of the distribution of excesses through a generalized probability-weighted moments method. J. Statist. Plann. Inference 137 (2007) 841–857. Google Scholar
Drees, H. and Kaufmann, E., Selecting the optimal sample fraction in univariate extreme value estimation. Stoc. Proc. Appl. 75 (1998) 149–172. Google Scholar
Fraga Alves, M.I., de Haan, L. and Lin, T., Estimation of the parameter controlling the speed of convergence in extreme value theory. Math. Methods Stat. 12 (2003) 155–176. Google Scholar
Fraga Alves, M.I., Gomes, M.I. and de Haan, L., A new class of semi-parametric estimators of the second order parameter. Portugaliae Mathematica 60 (2003) 193–213. Google Scholar
Fraga Alves, M.I., de Haan, L. and Lin, T., Third order extended regular variation. Publ. Inst. Math. 80 (2006) 109–120. Google Scholar
Fraga Alves, M.I., Gomes, M.I., de Haan, L. and Neves, C., A note on second order conditions in extreme value theory : linking general and heavy tail conditions. REVSTAT Stat. J. 5 (2007) 285–304. Google Scholar
Gomes, M.I. and Martins, J., “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Extremes 5 (2002) 5–31. Google Scholar
Gomes, M.I., de Haan, L. and Peng, L., Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes 5 (2002) 387–414. Google Scholar
Hall, P. and Welsh, A.H., Adaptive estimates of parameters of regular variation. Ann. Stat. 13 (1985) 331–341. Google Scholar
Hosking, J. and Wallis, J., Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29 (1987) 339–349. Google Scholar
Peng, L., Asymptotically unbiased estimator for the extreme value index. Statist. Prob. Lett. 38 (1998) 107–115. Google Scholar
Pickands, J. III, Statistical inference using extreme order statistics. Ann. Statist. 3 (1975) 119–131. Google Scholar
Raoult, J.P. and Worms, R., Rate of convergence for the generalized Pareto approximation of the excesses. Adv. Applied Prob. 35 (2003) 1007–1027. Google Scholar
R.J. Serfling, Approximation Theorems of Mathematical Statistics. Wiley & Son (1980).
A.W. van der Vaart, Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics (2000).
Worms, R., Penultimate approximation for the distribution of the excesses. ESAIM : PS 6 (2002) 21–31.Google Scholar