Advertisement

A Log Probability Weighted Moment Estimator of Extreme Quantiles

  • Frederico CaeiroEmail author
  • Dora Prata Gomes
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 136)

Abstract

In this paper we consider the semi-parametric estimation of extreme quantiles of a right heavy-tail model. We propose a new Probability Weighted Moment estimator for extreme quantiles, which is obtained from the estimators of the shape and scale parameters of the tail. Under a second-order regular variation condition on the tail, of the underlying distribution function, we deduce the non degenerate asymptotic behaviour of the estimators under study and present an asymptotic comparison at their optimal levels. In addition, the performance of the estimators is illustrated through an application to real data.

Keywords

Extreme quantile Extreme value index Log probability weighted moment Optimal level Statistics of extremes  

Notes

Acknowledgments

Research partially supported by FCT – Fundação para a Ciência e a Tecnologia, project UID/MAT/00297/2013 (CMA/UNL), EXTREMA, PTDC/MAT /101736/2008.

References

  1. 1.
    Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes. Theory and Applications. Wiley, New York (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Beirlant, J., Figueiredo, F., Gomes, M.I., Vandewalle, B.: Improved reduced-bias tail index and quantile estimators. J. Stat. Plan. Inference 138(6), 1851–1870 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Beirlant, J., Caeiro, F., Gomes, M.I.: An overview and open research topics in statistics of univariate extremes. Revstat 10(1), 1–31 (2012)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Caeiro, F., Gomes, M.I.: Computational validation of an adaptative choice of optimal sample fractions. In: 58th World Statistics Congress of the International Statistical Institute, Dublin, pp. 282–289 (2011)Google Scholar
  5. 5.
    Caeiro, F., Gomes, M.I.: Semi-parametric tail inference through probability-weighted moments. J. Stat. Plan. Inference 141, 937–950 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Caeiro, F., Gomes, M.I.: Asymptotic comparison at optimal levels of reduced-bias extreme value index estimators. Stat. Neerl. 65, 462–488 (2011)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Caeiro, F., Gomes, M.I.: A class of semi-parametric probability weighted moment estimators. In: Oliveira, P.E., da Graça Temido, M., Henriques, C., Vichi, M. (eds.) Recent Developments in Modeling and Applications in Statistics, pp. 139–147. Springer, Berlin (2013)CrossRefGoogle Scholar
  8. 8.
    Caeiro, F. and Gomes, M.I.: A semi-parametric estimator of a shape second order parameter. In: Pacheco, A., Santos, R., Rosário Oliveira, M. and Paulino, C.D. (eds.) New Advances in Statistical Modeling and Applications, Studies in Theoretical and Applied Statistics, pp. 137–144, Springer (2014)Google Scholar
  9. 9.
    Caeiro, F., Gomes, M.I., Henriques-Rodrigues, L.: Reduced-bias tail index estimators under a third order framework. Commun. Stat. Theory Methods 38(7), 1019–1040 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Caeiro, F., Gomes, M.I., Vandewalle, B.: Semi-parametric probability-weighted moments estimation revisited. Methodol. Comput. Appl. 16(1), 1–29 (2014)Google Scholar
  11. 11.
    Ciuperca, G., Mercadier, C.: Semi-parametric estimation for heavy tailed distributions. Extremes 13(1), 55–87 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    David, H., Nagaraja, H.N.: Order Statistics. Wiley, New York (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    de Haan, L., Peng, L.: Comparison of tail index estimators. Stat. Neerl. 52, 60–70 (1998)CrossRefzbMATHGoogle Scholar
  14. 14.
    de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer, New York (2006)CrossRefGoogle Scholar
  15. 15.
    Dekkers, A., de Haan, L.: Optimal sample fraction in extreme value estimation. J. Multivar. Anal. 47(2), 173–195 (1993)CrossRefzbMATHGoogle Scholar
  16. 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), 193–213 (2003)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Goegebeur, Y., Beirlant, J., de Wet, T.: Kernel estimators for the second order parameter in extreme value statistics. J. Stat. Plan. Inference 140, 2632–2652 (2010)CrossRefzbMATHGoogle Scholar
  18. 18.
    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), 5–31 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Gomes, M.I., Canto e Castro, L., Fraga Alves, M.I., Pestana, D.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), 3–34 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Greenwood, J.A., Landwehr, J.M., Matalas, N.C., Wallis, J.R.: Probability weighted moments: definition and relation to parameters of several distributions expressable in inverse form. Water Resour. Res. 15, 1049–1054 (1979)CrossRefGoogle Scholar
  21. 21.
    Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)CrossRefzbMATHGoogle Scholar
  22. 22.
    Hosking, J., Wallis, J.: Parameter and quantile estimation for the generalized pareto distribution. Technometrics 29(3), 339–349 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Weissman, I.: Estimation of parameters of large quantiles based on the \(k\) largest observations. J. Am. Stat. Assoc. 73, 812–815 (1978)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.CMA and FCTUniversidade Nova de LisboaCaparicaPortugal

Personalised recommendations