# Estimation of second order parameters using probability weighted moments

ESAIM: Probability and Statistics (2012)

- Volume: 16, page 97-113
- ISSN: 1292-8100

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topWorms, Julien, and Worms, Rym. "Estimation of second order parameters using probability weighted moments." ESAIM: Probability and Statistics 16 (2012): 97-113. <http://eudml.org/doc/273613>.

@article{Worms2012,

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.},

author = {Worms, Julien, Worms, Rym},

journal = {ESAIM: Probability and Statistics},

keywords = {extreme values; domain of attraction; excesses; generalized Pareto distribution; probability-weighted moments; second order parameter; third order condition; second-order parameter; third-order condition},

language = {eng},

pages = {97-113},

publisher = {EDP-Sciences},

title = {Estimation of second order parameters using probability weighted moments},

url = {http://eudml.org/doc/273613},

volume = {16},

year = {2012},

}

TY - JOUR

AU - Worms, Julien

AU - Worms, Rym

TI - Estimation of second order parameters using probability weighted moments

JO - ESAIM: Probability and Statistics

PY - 2012

PB - EDP-Sciences

VL - 16

SP - 97

EP - 113

AB - 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.

LA - eng

KW - extreme values; domain of attraction; excesses; generalized Pareto distribution; probability-weighted moments; second order parameter; third order condition; second-order parameter; third-order condition

UR - http://eudml.org/doc/273613

ER -

## References

top- [1] A. Balkema and L. de Haan, Residual life time at a great age. Ann. Probab.2 (1974) 792–801. Zbl0295.60014
- [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. Zbl1155.62037MR2493012
- [3] G. Ciuperca and C. Mercadier, Semi-parametric estimation for heavy tailed distributions. Extremes13 (2010) 55–87. Zbl1226.62053MR2593951
- [4] J. Diebolt, A. Guillou and R. Worms, Asymptotic behaviour of the probability-weighted moments and penultimate approximation. ESAIM : PS 7 (2003) 217–236. Zbl1017.60060MR1987787
- [5] J. Diebolt, A. Guillou and I. Rached, Approximation of the distribution of excesses through a generalized probability-weighted moments method. J. Statist. Plann. Inference137 (2007) 841–857. Zbl1107.60027MR2301720
- [6] J. Diebolt, A. Guillou and I. Rached, Approximation of the distribution of excesses through a generalized probability-weighted moments method. J. Statist. Plann. Inference137 (2007) 841–857. Zbl1107.60027MR2301720
- [7] H. Drees and E. Kaufmann, Selecting the optimal sample fraction in univariate extreme value estimation. Stoc. Proc. Appl.75 (1998) 149–172. Zbl0926.62013MR1632189
- [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. MR2025356
- [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 Mathematica60 (2003) 193–213. Zbl1042.62050MR1984031
- [10] M.I. Fraga Alves, L. de Haan and T. Lin, Third order extended regular variation. Publ. Inst. Math.80 (2006) 109–120. Zbl1164.26304MR2281909
- [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. Zbl1149.62040MR2365929
- [12] M.I. Gomes and J. Martins, “Asymptotically unbiased” estimators of the tail index based on external estimation of the second order parameter. Extremes5 (2002) 5–31. Zbl1037.62044MR1947785
- [13] M.I. Gomes, L. de Haan and L. Peng, Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes5 (2002) 387–414. Zbl1039.62027MR2002125
- [14] P. Hall and A.H. Welsh, Adaptive estimates of parameters of regular variation. Ann. Stat.13 (1985) 331–341. Zbl0605.62033MR773171
- [15] J. Hosking and J. Wallis, Parameter and quantile estimation for the generalized Pareto distribution. Technometrics29 (1987) 339–349. Zbl0628.62019MR906643
- [16] L. Peng, Asymptotically unbiased estimator for the extreme value index. Statist. Prob. Lett.38 (1998) 107–115. Zbl1246.62129MR1627906
- [17] J. Pickands III, Statistical inference using extreme order statistics. Ann. Statist.3 (1975) 119–131. Zbl0312.62038MR423667
- [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. Zbl1044.60041MR2014267
- [19] R.J. Serfling, Approximation Theorems of Mathematical Statistics. Wiley & Son (1980). Zbl1001.62005MR595165
- [20] A.W. van der Vaart, Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics (2000). Zbl0910.62001
- [21] R. Worms, Penultimate approximation for the distribution of the excesses. ESAIM : PS 6 (2002) 21–31. Zbl0992.60056MR1888136

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