Functional kernel estimators of conditional extreme quantiles

We address the estimation of "extreme" conditional quantiles i.e. when their order converges to one as the sample size increases. Conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed kernel estimators. A Weissman-type estimator and kernel estimators of the conditional tail-index are derived, permitting to estimate extreme conditional quantiles of arbitrary order.Comment: arXiv admin note: text overlap with arXiv:1107.226

A new location-scale model for conditional heavy-tailed distributions

International audienceWe are interested in a location-scale model for heavy-tailed distributions where the covariate is deterministic. We first address the nonparametric estimation of the location and scale functions and derive an estimator of the conditional extreme-value index. Second, new estimators of the extreme conditional quantiles are introduced. The asymptotic properties of the estimators are established under mild assumptions

Estimation de courbes de niveaux extrêmes pour des lois à queues lourdes

National audienceLe problème d'estimation des courbes de niveaux extrêmes est équivalent à l'étude des quantiles conditionnels quand l'ordre du quantile tend vers un. Nous montrons que sous certaines conditions, il est possible d'estimer de telles courbes au moyen d'un estimateur à noyau de la fonction de survie conditionnelle. En conséquence, ce résultat nous permet d'introduire deux versions lisses de l'estimateur de l'indice de queue conditionnel indispensable lorsque l'on veut extrapoler. Nous établissons la loi limite des estimateurs ainsi construits. Pour conclure, une illustration sur données simulées est présentée

On kernel smoothing for extremal quantile regression

Nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution of the response are long established in statistics. Attention has been, however, restricted to ordinary quantiles staying away from the tails of the conditional distribution. The purpose of this paper is to extend their asymptotic theory far enough into the tails. We focus on extremal quantile regression estimators of a response variable given a vector of covariates in the general setting, whether the conditional extreme-value index is positive, negative, or zero. Specifically, we elucidate their limit distributions when they are located in the range of the data or near and even beyond the sample boundary, under technical conditions that link the speed of convergence of their (intermediate or extreme) order with the oscillations of the quantile function and a von-Mises property of the conditional distribution. A simulation experiment and an illustration on real data were presented. The real data are the American electric data where the estimation of conditional extremes is found to be of genuine interest.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ466 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Estimation of the tail-index in a conditional location-scale family of heavy-tailed distributions

International audienceWe introduce a location-scale model for conditional heavy-tailed distributions when the covariate is deterministic. First, nonparametric estimators of the location and scale functions are introduced. Second, an estimator of the conditional extreme-value index is derived. The asymptotic properties of the estimators are established under mild assumptions and their finite sample properties are illustrated both on simulated and real data