pp 1–40 | Cite as

Bias-corrected estimation for conditional Pareto-type distributions with random right censoring

  • Yuri GoegebeurEmail author
  • Armelle Guillou
  • Jing Qin


We consider bias-reduced estimation of the extreme value index in conditional Pareto-type models with random covariates when the response variable is subject to random right censoring. The bias-correction is obtained by fitting the extended Pareto distribution locally to the relative excesses over a high threshold using the maximum likelihood method. Convergence in probability and asymptotic normality of the estimators are established under suitable assumptions. The finite sample behaviour is illustrated with a simulation experiment and the method is applied to two real datasets.


Pareto-type model Random covariate Random right censoring Local estimation Bias-reduction 

AMS 2000 Subject Classifications

62G08 62G20 62G32 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



This work was supported by a research grant (VKR023480) from VILLUM FONDEN. Computation/simulation for the work described in this paper was supported by the DeIC National HPC Centre, SDU. The authors sincerely thank the editor, associate editor and the referees for their helpful comments and suggestions that led to substantial improvement of the paper. The authors also take pleasure in thanking Gilles Stupfler for providing the code to compute the estimator proposed in Stupfler (2016).


  1. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.L.: Statistics of Extremes – Theory and Applications. Wiley (2004)Google Scholar
  2. Beirlant, J., Guillou, A., Dierckx, G., Fils-Villetard, A.: Estimation of the extreme value index and extreme quantiles under random censoring. Extremes 10, 151–174 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Beirlant, J., Joossens, E., Segers, J.: Second-order refined peaks-over-threshold modelling for heavy-tailed distributions. J. Stat. Planning Inference 139, 2800–2815 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Beirlant, J., Bardoutsos, A, de Wet, T., Gijbels, I.: Bias reduced tail estimation for censored Pareto type distributions. Statist. Probab. Lett. 109, 78–88 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Billingsley, P.: Probability and Measure. Wiley, New York (1995)zbMATHGoogle Scholar
  6. Daouia, A., Gardes, L., Girard, S., Lekina, A.: Kernel estimators of extreme level curves. TEST 20, 311–333 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Daouia, A., Gardes, L., Girard, S.: On kernel smoothing for extremal quantile regression. Bernoulli 19, 2557–2589 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. de Haan, L., Ferreira, A.: Extreme Value Theory: An Introduction. Springer (2006)Google Scholar
  9. 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, 1833–1855 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Dierckx, G., Goegebeur, Y., Guillou, A.: Local robust and asymptotically unbiased estimation of conditional Pareto-type tails. TEST 23, 330–355 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Drees, H.: On smooth statistical tail functionals. Scand. J. Stat. 25, 187–210 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Dutang, C., Goegebeur, Y., Guillou, A.: Robust and bias-corrected estimation of the coefficient of tail dependence. Insurance Math. Econom. 57, 46–57 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Dutang, C., Goegebeur, Y., Guillou, A.: Robust and bias-corrected estimation of the probability of extreme failure sets. Sankhya A 78, 52–86 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Einmahl, J.H.J., Fils-Villetard, A., Guillou, A.: Statistics of extremes under random censoring. Bernoulli 14, 207–227 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Escobar-Bach, M., Goegebeur, Y., Guillou, A.: Local estimation of the conditional stable tail dependence function. Scand. J. Stat. 45, 590–617 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Feuerverger, A., Hall, P.: Estimating a tail exponent by modelling departure from a Pareto distribution. Ann. Stat. 27, 760–781 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Frees, E., Valdez, E.: Understanding relationships using copulas. North American Actuarial J. 2, 1–25 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Goegebeur, Y., Guillou, A., Osmann, M.: A local moment type estimator for the extreme value index in regression with random covariates. Can. J. Stat. 42, 487–507 (2014a)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Goegebeur, Y., Guillou, A., Schorgen, A.: Nonparametric regression estimation of conditional tails - the random covariate case. Statistics 48, 732–755 (2014b)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Gomes, M.I., Martins, M.J.: Bias-reduction and explicit semi-parametric estimation of the tail index. J. Stat. Planning Inference 124, 361–378 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Gomes, M.I., Neves, M.M.: Estimation of the extreme value index for randomly censored data. Biomt. Lett. 48, 1–22 (2011)Google Scholar
  22. Hall, P.: On some simple estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B 44, 37–42 (1982)MathSciNetzbMATHGoogle Scholar
  23. Lawless, J.F.: Statistical Models and Methods for Lifetime Data. Wiley (2003)Google Scholar
  24. Lehmann, E.L., Casella, G.: Theory of Point Estimation. Springer (1998)Google Scholar
  25. Ndao, P., Diop, A., Dupuy, J.F.: Non parametric estimation of the conditional tail index and extreme quantiles under random censoring. Comput. Stat. Data Anal. 79, 63–79 (2014)CrossRefzbMATHGoogle Scholar
  26. Ndao, P., Diop, A., Dupuy, J.F.: Nonparametric estimation of the conditional extreme value index with random covariates and censoring. J. Stat. Planning Inference 168, 20–37 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Reynkens, T., Verbelen, R., Beirlant, J., Antonio, K.: Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions. Insurance Math. Econom. 77, 65–77 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Rodriguez, R.N.: A guide to the Burr type XII distributions. Biometrika 64, 129–134 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  29. Stupfler, G.: A moment estimator for the conditional extreme-value index. Electron. J. Stat. 7, 2298–2353 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. Stupfler, G.: Estimating the conditional extreme-value index under random right-censoring. J. Multivar. Anal. 144, 1–24 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Stupfler, G.: On the study of extremes with dependent random right-censoring. Extremes, to appear (2018)Google Scholar
  32. van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  33. Venables, W.N., Ripley, B.D.: Modern Applied Statistics with S, 4th edn. Springer (2002)Google Scholar
  34. Wang, H., Tsai, C.L.: Tail index regression. J. Am. Stat. Assoc. 104, 1233–1240 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  35. Weissman, I.: Estimation of parameters and large quantiles based on the k largest observations. J. Am. Stat. Assoc. 73, 812–815 (1978)MathSciNetzbMATHGoogle Scholar
  36. Worms, J., Worms, R.: New estimators of the extreme value index under random right censoring, for heavy-tailed distributions. Extremes 17, 337–358 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  37. Yao, Q.: Conditional Predictive Regions for Stochastic Processes. Technical report, Institute of Mathematics and Statistics, University of Kent at Canterbury (1999)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  2. 2.Institut Recherche Mathématique Avancée, UMR 7501Université de Strasbourg et CNRSStrasbourg cedexFrance

Personalised recommendations