Méthodes bayésiennes computationnelles pour l'estimation de quantiles extrêmes à partir de données environnementales

This thesis lies at the intersection of two research domains: extreme value statistics and Bayesian statistics. The main objective here is to use Bayesian methods for
the estimation of extreme quantiles, and in particular the return levels of environmental datasets.

The adoption of a Bayesian paradigm is motivated by various challenges associated with the estimation of extreme quantiles. Firstly, it allows for the direct consideration of different sources of uncertainty in a point estimator, for example by using the so-called predictive distributions. Secondly, it enables access to credible intervals to quantify the estimation error. Lastly, one aim is to provide insights into quantifying the limits of extrapolation, in other words, determining how far it is reasonable to extrapolate the tail of the distribution with a reasonable error for quantile estimation.

The first contribution of this thesis focuses on enhancing computational Bayesian methods through the reparameterization of extreme value models. The second contribution concerns the improvement of a convergence diagnostic for MCMC algorithms known as Gelman–Rubin diagnostic and denoted by R-hat. A new version denoted is proposed, based on a localized approach that diagnoses convergence issues on a specific quantile of the target distribution. The third contribution of the thesis consists of preliminary results regarding the tail behavior of different Bayesian estimators for finite sample sizes. The aim is to understand how these estimators behave in the tail, when taking into account the uncertainty associated with parameter estimation.Lastly, the final contribution of this thesis entails the application of the model and all the previous results to a series of environmental datasets.

Directeurs:

• Stéphane Girard (Inria Grenoble )
• Julyan Arbel (Inria Grenoble )

Raporteurs:

• Robin Ryder (Université Paris Dauphine )
• Clément Dombry (Université de Franche Comté )

Examinateurs:

• Antonio Canale (University of Padua )
• Anne-Catherine Favre (Grenoble INP )