Contributions à l'estimation de quantiles extrêmes. Applications à des données environnementales

Keywords :

• Statistique non-paramétrique
• Mesures de rique
• Quantiles extrêmes
• Lois à queue lourde
• Lois à queue de type Weibull
• Extreme-value theory
• Nonparametric statistics
• Risk measures
• Extreme quantile
• Heavy-tailed distributions
• Weibull tail-distributions

This thesis takes place in the context of extreme value statistics. It provides two main contributions to this subject area.

In the recent literature on extreme value statistics, a model on tail distributions which encompasses Pareto-type distributions
as well as Weibull tail-distributions has been introduced. The two main types of decreasing of the survival function are thus modeled. An estimator of extreme quantiles has been deduced from this model, but it depends on two unknown parameters, making it useless in practical situations. The first contribution of this thesis is to propose estimators of these parameters. Plugging our estimators in the previous extreme quantiles estimator allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a simulation study and on a real data set of exceedances of the Nidd river in the Yorkshire (England).

The second contribution of this thesis is the introduction and the estimation of a new risk measure, the so-called Conditional Tail Moment. It is defined as the moment of order a>0 of the loss distribution above the quantile of order p in (0,1) of the survival function. Estimating the Conditional Tail Moment permits to estimate all risk measures based on conditional moments
such as the Value-at-Risk, the Conditional Tail Expectation, the Conditional Value-at-Risk, the Conditional Tail Variance or the Conditional Tail Skewness. Here, we focus on the estimation of these risk measures in case of extreme losses i.e. when p converges to 0 when the size of the sample increases. It is moreover assumed that the loss distribution is heavy-tailed and depends on a covariate. The estimation method thus combines nonparametric kernel methods with extreme-value statistics. The asymptotic distribution of the estimators is established and their finite sample behavior is illustrated both on simulated data and on a real data set of daily rainfalls in the Cévennes-Vivarais region (France).

Directors:

• Mr Stéphane Girard (Chargé de Recherche - INRIA )
• Mr Laurent Gardes (Université de Strasbourg )

Raporteurs:

• Mr Clément Dombry (Université de Poitiers )
• Mme Irène Gijbels (Université Catholique de Louvain )

Examinators:

• Mme Clémentine Prieur (Professeur - Université Joseph Fourier )
• Mme Véronique Maume-Deschamps (I SFA -Université Lyon 1 )