Abstract
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.
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Beirlant, J., Goegebeur, Y.: Regression with response distributions of Pareto-type. Comput. Stat. Data Anal. 42, 595–619 (2003)
Beirlant, J., Goegebeur, Y.: Local polynomial maximum likelihood estimation for Pareto-type distributions. J. Multivar. Anal. 89, 97–118 (2004)
Beirlant, J., Dierckx, G., Guillou, A., Stǎricǎ, C.: On exponential representations of log-spacings of extreme order statistics. Extremes 5, 157–180 (2002)
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)
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)
Buishand, T.A., de Haan, L., Zhou, C.: On spatial extremes: with application to a rainfall problem. Ann. Appl. Stat. 2, 624–642 (2008)
Chavez-Demoulin, V., Davison, A.C.: Generalized additive modelling of sample extremes. J. R. Stat. Soc., Ser. C 54, 207–222 (2005)
Coles, S., Pericchi, L.R.: Anticipating catastrophes through extreme value modelling. Appl. Stat. 52, 405–416 (2003)
Coles, S., Tawn, J.: A Bayesian analysis of extreme rainfall data. Appl. Stat. 45, 463–478 (1996)
Coles, S., Pericchi, L.R., Sisson, S.: A fully probabilistic approach to extreme rainfall modeling. J. Hydrol. 273, 35–50 (2003)
Consul, P.C., Jain, G.C.: On the log-gamma distribution and its properties. Stat. Hefe 12(2), 100–106 (1971)
Cooley, D., Nychka, D., Naveau, P.: Bayesian spatial modeling of extreme precipitation return levels. J. Am. Stat. Assoc. 102, 824–840 (2007)
Davison, A.C., Ramesh, N.I.: Local likelihood smoothing of sample extremes. J. R. Stat. Soc., Ser. B 62, 191–208 (2000)
Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds. J. R. Stat. Soc., Ser. B 52, 393–442 (1990)
Dekkers, A., de Haan, L.: On the estimation of the extreme-value index and large quantile estimation. Ann. Stat. 17, 1795–1832 (1989)
Diebolt, J., Gardes, L., Girard, S., Guillou, A.: Bias-reduced extreme quantiles estimators of Weibull distributions. J. Stat. Plan. Inference 138, 1389–1401 (2008)
Falk, M., Hüsler, J., Reiss, R.D.: Laws of Small Numbers: Extremes and Rare Events, 2nd edn. Birkhäuser, Basel (2004)
Fawcett, L., Walshaw, D.: Improved estimation for temporally clustered extremes. Environmetrics 18, 173–188 (2007)
Gangopadhyay, A.K.: A note on the asymptotic behavior of conditional extremes. Stat. Probab. Lett. 25, 163–170 (1995)
Gardes, L., Girard, S.: Estimating extreme quantiles of Weibull tail-distributions. Commun. Stat. Theory Methods 34, 1065–1080 (2005)
Gardes L., Girard, S., Lekina, A.: Functional nonparametric estimation of conditional extreme quantiles. J. Multivar. Anal. 101, 419–433 (2010)
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)
Girard, S.: A Hill type estimate of the Weibull tail-coefficient. Commun. Stat. Theory Methods 33, 205–234 (2004)
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)
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)
Hall, P., Tajvidi, N.: Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data. Stat. Sci. 15, 153–167 (2000)
Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)
Kratz, M., Resnick, S.: The QQ-estimator and heavy tails. Stoch. Models 12, 699–724 (1996)
Loftsgaarden, D., Quesenberry, C.: A nonparametric estimate of a multivariate density function. Ann. Math. Stat. 36, 1049–1051 (1965)
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)
Padoan, S., Ribatet, M., Sisson, S.: Likelihood-based inference for max-stable processes. J. Am. Stat. Assoc. (2010, in press)
Schultze, J., Steinebach, J.: On least squares estimates of an exponential tail coefficient. Stat. Decis. 14, 353–372 (1996)
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)
Stone, C.: Consistent nonparametric regression. Ann. Stat. 5, 595–645 (1977)
Weissman, I.: Estimation of parameters and large quantiles based on the k largest observations. J. Am. Stat. Assoc. 73, 812–815 (1978)
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Gardes, L., Girard, S. Conditional extremes from heavy-tailed distributions: an application to the estimation of extreme rainfall return levels. Extremes 13, 177–204 (2010). https://doi.org/10.1007/s10687-010-0100-z
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DOI: https://doi.org/10.1007/s10687-010-0100-z