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 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

Estimation of extreme quantiles from heavy-tailed distributions in a semi-parametric location-dispersion regression model

National audienceThe modeling of extreme events arises in many fields such as finance, insurance or environmental science. A recurrent statistical problem is then the estimation of extreme quantiles associated with a random variable $Y$ recorded simultaneously with a multidimensional covariate x in R^d, the goal being to describe how tail characteristics such as extreme quantiles or small exceedance probabilities of the response variable Y may depend on the explanatory variable x. Here, we focus on the challenging situation where Y given x is heavy-tailed. Without additional assumptions on the pair (Y,x), the estimation of extreme conditional quantiles is addressed using semi-parametric method. More specifically, we assume that the response variable and the deterministic covariate are linked by a location-dispersion regression model Y=a(x)+b(x)Z where Z is a heavy-tailed random variable. This model is flexible since (i) no parametric assumptions are made on a(.), b(.) and Z, (ii) it allows for heteroscedasticity via the function b(.). Moreover, another feature of this model is that Y inherits its tail behaviour from Z which thus does not depend on the covariate x. We propose to take profit of this important property to decouple the estimation of the nonparametric and extreme structures. First, nonparametric estimators of the regression function a(.) and the dispersion function b(.) are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the residuals. A plug-in estimator of extreme conditional quantiles is then built using these two preliminary steps.We show that the resulting semi-parametric estimator is asymptotically Gaussian and may benefit from the same rate of convergence as in the unconditional situation. Its finite sample properties are illustrated both on simulated and real tsunami data

Estimation of extreme quantiles from heavy-tailed distributions in a location-dispersion regression model

International audienceWe consider a location-dispersion regression model for heavy-tailed distributions when the multidimensional covariate is deterministic. In a first step, nonparametric estimators of the regression and dispersion functions are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the residuals. Finally, a plug-in estimator of extreme conditional quantiles is built using these two preliminary steps. It is shown that the resulting semi-parametric estimator is asymptotically Gaussian and may benefit from the same rate of convergence as in the unconditional situation. Its finite sample properties are illustrated both on simulated and real tsunami data

Modélisation semi-paramétrique des extrêmes conditionnels

The main goal of this thesis is to propose new estimators of the tail-index as well as the conditional extreme quantiles in a family of heavy-tailed distributions. The considered family of distributions is defined from a regression model with a location function a(·) and a scale function b(·) which are unknown. The real random variable of interest Y is simultaneously recorded with a deterministic covariate x. The residuals Z of the model are independent of the covariate and their cumulative distribution function belongs to the Fréchet domain of attraction whose the tail-index γ is unknown and assumed to be constant. For more flexibility than purely parametric approaches, we opt for a semi-parametric estimation approach. Also, the constancy of the tail-index allows us to obtain, in the case of small samples, more reliable estimates than in certain purely non-parametric approaches existing in the literature. We establish the asymptotic properties of our estimators and present some results allowing to appreciate their finite sample properties both on simulated and real data.L'objectif principal de cette thèse est de proposer de nouveaux estimateurs de l’indice des valeurs extrêmes (indice de queue) ainsi que des quantiles extrêmes conditionnels pour une famille de distributions à queue lourde. La famille de distributions considérée est définie à partir d’un modèle de régression avec des paramètres fonctionnels de position a(·) et d’échelle b(·) inconnus. La variable d’intérêt Y, supposée aléatoire et réelle, est simultanément mesurée avec une covariable déterministe x. Les résidus Z du modèle sontindépendants de la covariable et sont distribués suivant une loi du domaine d’attraction de Fréchet d’indice de queue γ inconnu et supposé constant. Pour plus de souplesse que les approches purement paramétriques, nous préconisons une approche d’estimation semi-paramétrique. Aussi, la constance de l’indice de queue nous permet d’obtenir, dans le cas de petits échantillons, des estimations plus fiables que dans certaines approches purement non paramétriques existant dans la littérature. Nous établissons les propriétés asymptotiques de nos estimateurs et présentons, sur des simulations aussi bien que sur des données réelles, des résultats permettant d’apprécier leur comportement pour des échantillons de taille finie

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

We 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

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 of extreme quantiles from heavy-tailed distributions in a location-dispersion regression model

International audienceWe consider a location-dispersion regression model for heavy-tailed distributions when the multidimensional covariate is deterministic. In a first step, nonparametric estimators of the regression and dispersion functions are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the residuals. Finally, a plug-in estimator of extreme conditional quantiles is built using these two preliminary steps. It is shown that the resulting semi-parametric estimator is asymptotically Gaussian and may benefit from the same rate of convergence as in the unconditional situation. Its finite sample properties are illustrated both on simulated and real tsunami data

Estimation of extreme quantiles from heavy-tailed distributions in a semi-parametric location-dispersion regression model

National audienceThe modeling of extreme events arises in many fields such as finance, insurance or environmental science. A recurrent statistical problem is then the estimation of extreme quantiles associated with a random variable $Y$ recorded simultaneously with a multidimensional covariate x in R^d, the goal being to describe how tail characteristics such as extreme quantiles or small exceedance probabilities of the response variable Y may depend on the explanatory variable x. Here, we focus on the challenging situation where Y given x is heavy-tailed. Without additional assumptions on the pair (Y,x), the estimation of extreme conditional quantiles is addressed using semi-parametric method. More specifically, we assume that the response variable and the deterministic covariate are linked by a location-dispersion regression model Y=a(x)+b(x)Z where Z is a heavy-tailed random variable. This model is flexible since (i) no parametric assumptions are made on a(.), b(.) and Z, (ii) it allows for heteroscedasticity via the function b(.). Moreover, another feature of this model is that Y inherits its tail behaviour from Z which thus does not depend on the covariate x. We propose to take profit of this important property to decouple the estimation of the nonparametric and extreme structures. First, nonparametric estimators of the regression function a(.) and the dispersion function b(.) are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the residuals. A plug-in estimator of extreme conditional quantiles is then built using these two preliminary steps.We show that the resulting semi-parametric estimator is asymptotically Gaussian and may benefit from the same rate of convergence as in the unconditional situation. Its finite sample properties are illustrated both on simulated and real tsunami data

Estimation of extreme quantiles from heavy-tailed distributions in a location-dispersion regression model

International audienceWe consider a location-dispersion regression model for heavy-tailed distributions when the multidimensional covariate is deterministic. In a first step, nonparametric estimators of the regression and dispersion functions are introduced. This permits, in a second step, to derive an estimator of the conditional extreme-value index computed on the residuals. Finally, a plug-in estimator of extreme conditional quantiles is built using these two preliminary steps. It is shown that the resulting semi-parametric estimator is asymptotically Gaussian and may benefit from the same rate of convergence as in the unconditional situation. Its finite sample properties are illustrated both on simulated and real tsunami data