Estimation of second order parameters using probability weighted moments
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 97-113.

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.

DOI : 10.1051/ps/2010017
Classification : 62G32, 60G70
Mots-clés : extreme values, domain of attraction, excesses, generalized Pareto distribution, probability-weighted moments, second order parameter, third order condition
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     author = {Worms, Julien and Worms, Rym},
     title = {Estimation of second order parameters using probability weighted moments},
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     pages = {97--113},
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     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2010017/}
}
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Worms, Julien; Worms, Rym. Estimation of second order parameters using probability weighted moments. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 97-113. doi : 10.1051/ps/2010017. http://archive.numdam.org/articles/10.1051/ps/2010017/

[1] A. Balkema and L. De Haan, Residual life time at a great age. Ann. Probab. 2 (1974) 792-801. | Zbl

[2] F. Caeiro, M.I. Gomes and D. Pestana, A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator. Stat. Probab. Lett. 79 (2009) 295-303. | MR | Zbl

[3] G. Ciuperca and C. Mercadier, Semi-parametric estimation for heavy tailed distributions. Extremes 13 (2010) 55-87. | MR | Zbl

[4] J. Diebolt, A. Guillou and R. Worms, Asymptotic behaviour of the probability-weighted moments and penultimate approximation. ESAIM : PS 7 (2003) 217-236. | Numdam | MR | Zbl

[5] J. Diebolt, A. Guillou and I. Rached, Approximation of the distribution of excesses through a generalized probability-weighted moments method. J. Statist. Plann. Inference 137 (2007) 841-857. | MR | Zbl

[6] J. Diebolt, A. Guillou and I. Rached, Approximation of the distribution of excesses through a generalized probability-weighted moments method. J. Statist. Plann. Inference 137 (2007) 841-857. | MR | Zbl

[7] H. Drees and E. Kaufmann, Selecting the optimal sample fraction in univariate extreme value estimation. Stoc. Proc. Appl. 75 (1998) 149-172. | MR | Zbl

[8] M.I. Fraga Alves, L. De Haan and T. Lin, Estimation of the parameter controlling the speed of convergence in extreme value theory. Math. Methods Stat. 12 (2003) 155-176. | MR

[9] M.I. Fraga Alves, M.I. Gomes and L. De Haan, A new class of semi-parametric estimators of the second order parameter. Portugaliae Mathematica 60 (2003) 193-213. | MR | Zbl

[10] M.I. Fraga Alves, L. De Haan and T. Lin, Third order extended regular variation. Publ. Inst. Math. 80 (2006) 109-120. | MR | Zbl

[11] M.I. Fraga Alves, M.I. Gomes, L. De Haan and C. Neves, A note on second order conditions in extreme value theory : linking general and heavy tail conditions. REVSTAT Stat. J. 5 (2007) 285-304. | MR | Zbl

[12] M.I. Gomes and J. Martins, “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Extremes 5 (2002) 5-31. | MR | Zbl

[13] M.I. Gomes, L. De Haan and L. Peng, Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes 5 (2002) 387-414. | MR | Zbl

[14] P. Hall and A.H. Welsh, Adaptive estimates of parameters of regular variation. Ann. Stat. 13 (1985) 331-341. | MR | Zbl

[15] J. Hosking and J. Wallis, Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29 (1987) 339-349. | MR | Zbl

[16] L. Peng, Asymptotically unbiased estimator for the extreme value index. Statist. Prob. Lett. 38 (1998) 107-115. | MR | Zbl

[17] J. Pickands Iii, Statistical inference using extreme order statistics. Ann. Statist. 3 (1975) 119-131. | MR | Zbl

[18] J.P. Raoult and R. Worms, Rate of convergence for the generalized Pareto approximation of the excesses. Adv. Applied Prob. 35 (2003) 1007-1027. | MR | Zbl

[19] R.J. Serfling, Approximation Theorems of Mathematical Statistics. Wiley & Son (1980). | MR | Zbl

[20] A.W. Van Der Vaart, Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics (2000). | Zbl

[21] R. Worms, Penultimate approximation for the distribution of the excesses. ESAIM : PS 6 (2002) 21-31. | Numdam | MR | Zbl

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