Robust conditional Weibull-type estimation

  • Yuri GoegebeurEmail author
  • Armelle Guillou
  • Théo Rietsch


We study nonparametric robust tail coefficient estimation when the variable of interest, assumed to be of Weibull type, is observed simultaneously with a random covariate. In particular, we introduce a robust estimator for the tail coefficient, using the idea of the density power divergence, based on the relative excesses above a high threshold. The main asymptotic properties of our estimator are established under very general assumptions. The finite sample performance of the proposed procedure is evaluated by a small simulation experiment.


Weibull-type distribution Tail coefficient Density power divergence Local estimation 


  1. Basu, A., Harris, I. R., Hjort, N. L., Jones, M. C. (1998). Robust and efficient estimation by minimizing a density power divergence. Biometrika, 85, 549–559.Google Scholar
  2. Beirlant, J., Broniatowski, M., Teugels, J. L., Vynckier, P. (1995). The mean residual life function at great age: applications to tail estimation. Journal of Statistical Planning and Inference, 45, 21–48.Google Scholar
  3. Billingsley, P. (1995). Probability and measure. Wiley series in probability and mathematical statistics. New York: Wiley.Google Scholar
  4. Brazauskas, V., Serfling, R. (2000). Robust estimation of tail parameters for two-parameter Pareto and exponential models via generalized quantile statistics. Extremes, 3, 231–249.Google Scholar
  5. Broniatowski, M. (1993). On the estimation of the Weibull tail coefficient. Journal of Statistical Planning and Inference, 35, 349–366.CrossRefzbMATHMathSciNetGoogle Scholar
  6. Daouia, A., Gardes, L., Girard, S., Lekina, A. (2011). Kernel estimators of extreme level curves. Test, 20, 311–333.Google Scholar
  7. Daouia, A., Gardes, L., Girard, S. (2013). On kernel smoothing for extremal quantile regression. Bernoulli, 19, 2557–2589.Google Scholar
  8. de Haan, L., Ferreira, A. (2006). Extreme value theory: an introduction. New York: Springer.Google Scholar
  9. de Wet, T., Goegebeur, Y., Guillou, A. Osmann, M. (2013). Kernel regression with Weibull-type tails. Submitted.Google Scholar
  10. Diebolt, J., Gardes, L., Girard, S., Guillou, A. (2008). Bias-reduced estimators of the Weibull tail-coefficient. Test, 17, 311–331.Google Scholar
  11. Dierckx, G., Beirlant, J., De Waal, D., Guillou, A. (2009). A new estimation method for Weibull-type tails based on the mean excess function. Journal of Statistical Planning and Inference, 139, 1905–1920.Google Scholar
  12. Dierckx, G., Goegebeur, Y., Guillou, A. (2013). An asymptotically unbiased minimum density power divergence estimator for the Pareto-tail index. Journal of Multivariate Analysis, 121, 70–86.Google Scholar
  13. Dierckx, G., Goegebeur, Y., Guillou, A. (2014). Local robust and asymptotically unbiased estimation of conditional Pareto-type tails. Test. doi: 10.1007/s11749-013-0350-6
  14. Dupuis, D., Field, C. (1998). Robust estimation of extremes. Canadian Journal of Statistics, 26, 119–215.Google Scholar
  15. Gannoun, A., Girard, S., Guinot, C., Saracco, J. (2002). Reference ranges based on nonparametric quantile regression. Statistics in Medicine, 21, 3119–3135.Google Scholar
  16. Gardes, L., Girard, S. (2005). Estimating extreme quantiles of Weibull tail distributions. Communications in Statistics-Theory and Methods, 34, 1065–1080.Google Scholar
  17. Gardes, L., Girard, S. (2008a). A moving window approach for nonparametric estimation of the conditional tail index. Journal of Multivariate Analysis, 99, 2368–2388.Google Scholar
  18. Gardes, L., Girard, S. (2008b). Estimation of the Weibull-tail coefficient with linear combination of upper order statistics. Journal of Statistical Planning and Inference, 138, 1416–1427.Google Scholar
  19. Gardes, L., Stupfler, G. (2013). Estimation of the conditional tail index using a smoothed local Hill estimator. Extremes. doi: 10.1007/s10687-013-0174-5
  20. Gardes, L., Girard, S., Lekina, A. (2010). Functional nonparametric estimation of conditional extreme quantiles. Journal of Multivariate Analysis, 101, 419–433.Google Scholar
  21. Geluk, J.L., de Haan, L. (1987). Regular variation, extensions and Tauberian theorems. CWI Tract 40. Amsterdam: Center for Mathematics and Computer Science.Google Scholar
  22. Girard, S. (2004). A Hill type estimator of the Weibull tail coefficient. Communications in Statistics-Theory and Methods, 33, 205–234.CrossRefzbMATHMathSciNetGoogle Scholar
  23. Goegebeur, Y., Guillou, A. (2011). A weighted mean excess function approach to the estimation of Weibull-type tails. Test, 20, 138–162.Google Scholar
  24. Goegebeur, Y., Beirlant, J., de Wet, T. (2010). Generalized kernel estimators for the Weibull tail coefficient. Communications in Statistics-Theory and Methods, 39, 3695–3716.Google Scholar
  25. Goegebeur, Y., Guillou, A., Schorgen, A. (2013a). Nonparametric regression estimation of conditional tails–the random covariate case. Statistics. doi: 10.1080/02331888.2013.800064
  26. Goegebeur, Y., Guillou, A., Stupfler, G. (2013b). Uniform asymptotic properties of a nonparametric regression estimator of conditional tails. Submitted.
  27. Hall, P. (1982). On some simple estimates of an exponent of regular variation. Journal of the Royal Statistical Society, Series B: Statistical Methodology, 44, 37–42.zbMATHGoogle Scholar
  28. Juárez, S., Schucany, W. (2004). Robust and efficient estimation for the generalized Pareto distribution. Extremes, 7, 237–251.Google Scholar
  29. Kim, M., Lee, S. (2008). Estimation of a tail index based on minimum density power divergence. Journal of Multivariate Analysis, 99, 2453–2471.Google Scholar
  30. Klüppelberg, C., Villaseñor, J. A. (1993). Estimation of distribution tails–a semiparametric approach. Blätter der Deutschen Gesellschaft für Versicherungsmathematik, 21, 213–235.Google Scholar
  31. Lehmann, E. L., Casella, G. (1998). Theory of point estimation. New York: Springer.Google Scholar
  32. Parzen, E. (1962). On estimation of a probability density function and mode. Annals of Mathematical Statistics, 33, 1065–1076.CrossRefzbMATHMathSciNetGoogle Scholar
  33. Peng, L., Welsh, A. (2001). Robust estimation of the generalized Pareto distribution. Extremes, 4, 53–65.Google Scholar
  34. Severini, T. (2005). Elements of distribution theory. Cambridge series in statistical and probabilistic mathematics. New York: Cambridge University Press.Google Scholar
  35. Vandewalle, B., Beirlant, J., Christmann, A., Hubert, M. (2007). A robust estimator for the tail index of Pareto-type distributions. Computational Statistics and Data Analysis, 51, 6252–6268.Google Scholar
  36. Wang, H., Tsai, C. L. (2009). Tail index regression. Journal of the American Statistical Association, 104, 1233–1240.Google Scholar
  37. Yao, Q. (1999). Conditional predictive regions for stochastic processes. Technical report, University of Kent at Canterbury.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2014

Authors and Affiliations

  • Yuri Goegebeur
    • 1
    Email author
  • Armelle Guillou
    • 2
  • Théo Rietsch
    • 2
    • 3
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark
  2. 2.Institut Recherche Mathématique Avancée, UMR 7501Université de Strasbourg et CNRSStrasbourg CedexFrance
  3. 3.Laboratoire des Sciences du Climat et de l’Environnement, LSCE-IPSL-CNRSGif-sur-YvetteFrance

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