, Volume 13, Issue 2, pp 177–204 | Cite as

Conditional extremes from heavy-tailed distributions: an application to the estimation of extreme rainfall return levels

  • Laurent Gardes
  • Stéphane Girard


This paper is dedicated to the estimation of extreme quantiles and the tail index from heavy-tailed distributions when a covariate is recorded simultaneously with the quantity of interest. A nearest neighbor approach is used to construct our estimators. Their asymptotic normality is established under mild regularity conditions and their finite sample properties are illustrated on a simulation study. An application to the estimation of pointwise return levels of extreme rainfalls in the Cévennes-Vivarais region is provided.


Conditional extreme quantiles Heavy-tailed distribution Nearest neighbor estimator Extreme rainfalls 

AMS 2000 Subject Classifications

62G32 62G05 62E20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Beirlant, J., Goegebeur, Y.: Regression with response distributions of Pareto-type. Comput. Stat. Data Anal. 42, 595–619 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  2. Beirlant, J., Goegebeur, Y.: Local polynomial maximum likelihood estimation for Pareto-type distributions. J. Multivar. Anal. 89, 97–118 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. Beirlant, J., Dierckx, G., Guillou, A., Stǎricǎ, C.: On exponential representations of log-spacings of extreme order statistics. Extremes 5, 157–180 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. In: Doran, R., Flajolet, P., Ismail, M., Lam, T.-Y., Lutwak, E., Rota, G.C. (eds) Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1987)Google Scholar
  5. Bois, P., Obled, C., de Saintignon, M.F., Mailloux, H.: Atlas Expérimental des Risques de Pluies Intenses Cévennes - Vivarais, 2ème édn. Pôle grenoblois d’études et de recherche pour la prévention des risques naturels, Grenoble (1997)Google Scholar
  6. Buishand, T.A., de Haan, L., Zhou, C.: On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat. 2, 624–642 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  7. Chavez-Demoulin, V., Davison, A.C.: Generalized additive modelling of sample extremes. J. R. Stat. Soc., Ser. C 54, 207–222 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  8. Coles, S., Pericchi, L.R.: Anticipating catastrophes through extreme value modelling. Appl. Stat. 52, 405–416 (2003)zbMATHMathSciNetGoogle Scholar
  9. Coles, S., Tawn, J.: A Bayesian analysis of extreme rainfall data. Appl. Stat. 45, 463–478 (1996)CrossRefGoogle Scholar
  10. Coles, S., Pericchi, L.R., Sisson, S.: A fully probabilistic approach to extreme rainfall modeling. J. Hydrol. 273, 35–50 (2003)CrossRefGoogle Scholar
  11. Consul, P.C., Jain, G.C.: On the log-gamma distribution and its properties. Stat. Hefe 12(2), 100–106 (1971)CrossRefGoogle Scholar
  12. Cooley, D., Nychka, D., Naveau, P.: Bayesian spatial modeling of extreme precipitation return levels. J. Am. Stat. Assoc. 102, 824–840 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. Davison, A.C., Ramesh, N.I.: Local likelihood smoothing of sample extremes. J. R. Stat. Soc., Ser. B 62, 191–208 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds. J. R. Stat. Soc., Ser. B 52, 393–442 (1990)zbMATHMathSciNetGoogle Scholar
  15. Dekkers, A., de Haan, L.: On the estimation of the extreme-value index and large quantile estimation. Ann. Stat. 17, 1795–1832 (1989)zbMATHCrossRefGoogle Scholar
  16. Diebolt, J., Gardes, L., Girard, S., Guillou, A.: Bias-reduced extreme quantiles estimators of Weibull distributions. J. Stat. Plan. Inference 138, 1389–1401 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  17. Falk, M., Hüsler, J., Reiss, R.D.: Laws of Small Numbers: Extremes and Rare Events, 2nd edn. Birkhäuser, Basel (2004)zbMATHGoogle Scholar
  18. Fawcett, L., Walshaw, D.: Improved estimation for temporally clustered extremes. Environmetrics 18, 173–188 (2007)CrossRefMathSciNetGoogle Scholar
  19. Gangopadhyay, A.K.: A note on the asymptotic behavior of conditional extremes. Stat. Probab. Lett. 25, 163–170 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  20. Gardes, L., Girard, S.: Estimating extreme quantiles of Weibull tail-distributions. Commun. Stat. Theory Methods 34, 1065–1080 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  21. Gardes L., Girard, S., Lekina, A.: Functional nonparametric estimation of conditional extreme quantiles. J. Multivar. Anal. 101, 419–433 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  22. Geluk, J.L., de Haan, L.: Regular variation, extensions and Tauberian theorems. In: Math Centre Tracts, vol. 40. Centre for Mathematics and Computer Science, Amsterdam (1987)Google Scholar
  23. Girard, S.: A Hill type estimate of the Weibull tail-coefficient. Commun. Stat. Theory Methods 33, 205–234 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  24. Gomes, M.I., de Haan, L., Peng, L.: Semi-parametric estimation of the second order parameter in statistics of extremes. Extremes 5, 387–414 (2003)zbMATHCrossRefGoogle Scholar
  25. Gomes, M.I., Caeiro, F., Figueiredo, F.: Bias reduction of a tail index estimator through an external estimation of the second-order parameter. Statistics 38, 497–510 (2004)zbMATHMathSciNetGoogle Scholar
  26. Hall, P., Tajvidi, N.: Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data. Stat. Sci. 15, 153–167 (2000)CrossRefMathSciNetGoogle Scholar
  27. Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)zbMATHCrossRefGoogle Scholar
  28. Kratz, M., Resnick, S.: The QQ-estimator and heavy tails. Stoch. Models 12, 699–724 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  29. Loftsgaarden, D., Quesenberry, C.: A nonparametric estimate of a multivariate density function. Ann. Math. Stat. 36, 1049–1051 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  30. Molinié, G., Yates, E., Ceresetti, D., Anquetin, S., Boudevillain, B., Creutin, J.D., Bois, P.: Rainfall regimes in a mountainous Mediterranean region: statistical analysis at short time steps (2010, technical report)Google Scholar
  31. Padoan, S., Ribatet, M., Sisson, S.: Likelihood-based inference for max-stable processes. J. Am. Stat. Assoc. (2010, in press)Google Scholar
  32. Schultze, J., Steinebach, J.: On least squares estimates of an exponential tail coefficient. Stat. Decis. 14, 353–372 (1996)zbMATHMathSciNetGoogle Scholar
  33. Smith, R.L.: Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone (with discussion). Stat. Sci. 4, 367–393 (1989)zbMATHCrossRefGoogle Scholar
  34. Stone, C.: Consistent nonparametric regression. Ann. Stat. 5, 595–645 (1977)zbMATHCrossRefGoogle Scholar
  35. Weissman, I.: Estimation of parameters and large quantiles based on the k largest observations. J. Am. Stat. Assoc. 73, 812–815 (1978)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Team MistisINRIA Rhône-Alpes and Laboratoire Jean KuntzmannSaint-Ismier CedexFrance

Personalised recommendations