Adaptive estimation of heavy right tails: resampling-based methods in action

详细信息    查看全文

• 作者：M. Ivette Gomes (1)
  Fernanda Figueiredo (2)
  M. Manuela Neves (3)
  
• 关键词：Statistics of extremes ; Semi ; parametric estimation ; Resampling ; based methodology ; Primary-2G32 ; 62E20 ; Secondary-5C05
• 刊名：Extremes
• 出版年：2012
• 出版时间：December 2012
• 年：2012
• 卷：15
• 期：4
• 页码：463-489
• 参考文献：1. Beirlant, J., Vynckier, P., Teugels, J.: Excess functions and estimation of the extreme-value index. Bernoulli 2, 293-18 (1996) 10.2307/3318416">CrossRef
  2. Beirlant, J., Dierckx, G., Goegebeur, Y., Matthys, G.: Tail index estimation and an exponential regression model. Extremes 2, 177-00 (1999) 10.1023/A:1009975020370">CrossRef
  3. Beirlant, J., Dierckx, G., Guillou, A.: Estimation of the extreme-value index and generalized quantile plots. Bernoulli 11(6), 949-70 (2005) 10.3150/bj/1137421635">CrossRef
  4. Caeiro, F., Gomes, M.I.: A new class of estimators of a “scale-second order parameter. Extremes 9, 193-11 (2006) 10.1007/s10687-006-0026-7">CrossRef
  5. Caeiro, F., Gomes, M.I.: Asymptotic comparison at optimal levels of reduced-bias extreme value index estimators. Stat. Neerl. 65(4), 462-88 10.1111/j.1467-9574.2011.00495.x">CrossRef
  6. Caeiro, F., Gomes, M.I., Pestana, D.: Direct reduction of bias of the classical Hill estimator. Revstat 3(2), 111-36 (2005)
  7. Caeiro, F., Gomes, M.I., Henriques-Rodrigues, L.: Reduced-bias tail index estimators under a third order framework. Commun. Stat., Theory Methods 38(7), 1019-040 (2009) 10.1080/03610920802361415">CrossRef
  8. Ciuperca, G., Mercadier, C.: Semi-parametric estimation for heavy tailed distributions. Extremes 13(1), 55-7 (2010) 10.1007/s10687-009-0086-6">CrossRef
  9. Danielsson, J., de Haan, L., Peng, L., de Vries, C.G.: Using a bootstrap method to choose the sample fraction in the tail index estimation. J. Multivar. Anal. 76, 226-48 (2001) 10.1006/jmva.2000.1903">CrossRef
  10. de Haan, L., Ferreira, A.: / Extreme Value Theory: An Introduction. Springer, New York (2006)
  11. de Haan, L., Peng, L.: Comparison of tail index estimators. Stat. Neerl. 52, 60-0 (1998) 10.1111/1467-9574.00068">CrossRef
  12. Dekkers, A.L.M., Einmahl, J.H.J., de Haan, L.: A moment estimator for the index of an extreme-value distribution. Ann. Stat. 17, 1833-855 (1989) 10.1214/aos/1176347397">CrossRef
  13. Draisma, G., de Haan, L., Peng, L., Pereira, M.T.: A bootstrap-based method to achieve optimality in estimating the extreme value index. Extremes 2(4), 367-04 (1999) 10.1023/A:1009900215680">CrossRef
  14. Drees, H.: A general class of estimators of the extreme value index. J. Stat. Plan. Inference 98, 95-12 (1998) 10.1016/S0378-3758(97)00076-1">CrossRef
  15. Feuerverger, A., Hall, P.: Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Stat. 27, 760-81 (1999) 10.1214/aos/1018031215">CrossRef
  16. Fraga Alves, M.I., Gomes M.I., de Haan, L.: A new class of semi-parametric estimators of the second order parameter. Port. Math. 60(2), 194-13 (2003)
  17. Geluk, J., de Haan, L.: Regular Variation, Extensions and Tauberian Theorems. CWI Tract 40, Center for Mathematics and Computer Science, Amsterdam, Netherlands (1987)
  18. Goegebeur, Y., Beirlant, J., de Wet, T.: Linking Pareto-tail kernel goodness-of-fit statistics with tail index at optimal threshold and second order estimation. Revstat 6(1), 51-9 (2008)
  19. Goegebeur, Y., Beirlant, J., de Wet, T.: Kernel estimators for the second order parameter in extreme value statistics. J. Stat. Plan. Inference 140(9), 2632-652 (2010) 10.1016/j.jspi.2010.03.029">CrossRef
  20. Gomes, M.I., Martins, M.J.: “Asymptotically unbiased-estimators of the tail index based on external estimation of the second order parameter. Extremes 5(1), 5-1 (2002) 10.1023/A:1020925908039">CrossRef
  21. Gomes, M.I., Neves, C.: Asymptotic comparison of the mixed moment and classical extreme value index estimators. Stat. Probab. Lett. 78(6), 643-53 (2008) 10.1016/j.spl.2007.07.026">CrossRef
  22. Gomes, M.I., Oliveira, O.: The bootstrap methodology in Statistical Extremes—choice of the optimal sample fraction. Extremes 4(4), 331-58 (2001) 10.1023/A:1016592028871">CrossRef
  23. Gomes, M.I., Pestana, D.: A simple second order reduced-bias-tail index estimator. J. Stat. Comput. Simul. 77(6), 487-04 (2007a) 10.1080/10629360500282239">CrossRef
  24. Gomes, M.I., Pestana, D.: A sturdy reduced-bias extreme quantile (VaR) estimator. J. Am. Stat. Assoc. 102(477), 280-92 (2007b) 10.1198/016214506000000799">CrossRef
  25. Gomes, M.I., Martins, M.J., Neves, M.M.: Alternatives to a semi-parametric estimator of parameters of rare events—the Jackknife methodology. Extremes 3(3), 207-29 (2000) 10.1023/A:1011470010228">CrossRef
  26. Gomes, M.I., Martins, M.J., Neves, M.M.: Improving second order reduced-bias tail index estimation. Revstat 5(2), 177-07 (2007)
  27. Gomes, M.I., Canto e Castro, L., Fraga Alves, M.I., Pestana, D.: Statistics of extremes for iid data and breakthroughs in the estimation of the extreme value index: Laurens de Haan leading contributions. Extremes 11(1), 3-4 (2008a) 10.1007/s10687-007-0048-9">CrossRef
  28. Gomes, M.I., Fraga Alves, M.I., Araújo Santos, P.: PORT Hill and moment estimators for heavy-tailed models. Commun. Stat., Simul. Comput. 37, 1281-306 (2008b) 10.1080/03610910802050910">CrossRef
  29. Gomes, M.I., de Haan, L., Henriques-Rodrigues, L.: Tail index estimation for heavy-tailed models: accommodation of bias in weighted log-excesses. J. R. Stat. Soc. B70(1), 31-2 (2008c)
  30. Gomes, M.I., Pestana, D., Caeiro, F.: A note on the asymptotic variance at optimal levels of a bias-corrected Hill estimator. Stat. Probab. Lett. 79, 295-03 (2008d) 10.1016/j.spl.2008.08.016">CrossRef
  31. Gomes, M.I., Henriques-Rodrigues, L., Pereira, H., Pestana, D.: Tail index and second order parameters-semi-parametric estimation based on the log-excesses. J. Stat. Comput. Simul. 80(6), 653-66 (2010) 10.1080/00949650902755178">CrossRef
  32. Gomes, M.I., Mendon?a, S., Pestana, D.: Adaptive reduced-bias tail index and VaR estimation via the bootstrap methodology. Commun. Stat., Theory Methods 40(16), 2946-968 (2011) 10.1080/03610926.2011.562782">CrossRef
  33. Hall, P., Welsh, A.W.: Adaptive estimates of parameters of regular variation. Ann. Stat. 13, 331-41 (1985) 10.1214/aos/1176346596">CrossRef
  34. Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163-174 (1975) 10.1214/aos/1176343247">CrossRef
  35. Peng, L., Qi, Y.: Estimating the first and second order parameters of a heavy tailed distribution. Aust. N. Z. J. Stat. 46, 305-12 (2004) 10.1111/j.1467-842X.2004.00331.x">CrossRef
  36. Reiss, R.-D. Thomas, M.: Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, 3rd edn. Birkhaüser Verlag, Basel-Boston-Berlin (2007)
  
• 作者单位：M. Ivette Gomes (1)
  Fernanda Figueiredo (2)
  M. Manuela Neves (3)
  
  1. Universidade de Lisboa, FCUL, DEIO and CEAUL, Lisboa, Portugal
  2. Faculdade de Economia and CEAUL, Universidade do Porto, Porto, Portugal
  3. Instituto Superior de Agronomia and CEAUL, Universidade Técnica de Lisboa, Lisboa, Portugal
  
• ISSN：1572-915X

文摘

In this paper, we discuss an algorithm for the adaptive estimation of a positive extreme value index, γ, the primary parameter in Statistics of Extremes. Apart from the classical extreme value index estimators, we suggest the consideration of associated second-order corrected-bias estimators, and propose the use of resampling-based computer-intensive methods for an asymptotically consistent choice of the thresholds to use in the adaptive estimation of γ. The algorithm is described for a classical γ-estimator and associated corrected-bias estimator, but it can work similarly for the estimation of other parameters of extreme events, like a high quantile, the probability of exceedance or the return period of a high level.