Estimation bayesienne nonparametrique de copules

This thesis is based on two different projects that I have worked on during the last two years. These projects lead to two papers, the first one is accepted for publication in the Canadian Journal of Statistics and the second one should be submitted soon.;First Article. A bivariate distribution with continuous margins can be decomposed via a copula and its marginal distributions. In the case of extreme value distributions, the copula is characterized by a dependence function while each of the margins depends on three parameters. In this article we propose a Bayesian approach for the simultaneous estimation of the dependence function and the parameters defining the marginal distributions. A nonparametric model is constructed for the dependence function and a reversible jump MCMC algorithm is proposed for numerical evaluations of the Bayesian estimator. Comparisons are made with classical nonparametric estimators through an extensive simulation. We observe that our estimator outperforms every competitor in terms of mean integrated squared error, especially for small sample sizes, a much desirable quality when working with extreme values. Finally, we end this article by illustrating our estimation method on a hydrological data set and we make predictions based on the predictive distribution using a reversible jump MCMC algorithm.;Second Article. A bivariate distribution with continuous margins can be decomposed via a copula and its marginal distributions. On the space of copula functions, we construct a finite dimensional approximation subspace which is parameterized by a doubly stochastic matrix. The estimation is done in the space of doubly stochastic matrices also known as the Birkhoff polytope. The model enables the construction of estimators which belong to the class of copulas. A Bayesian approach is undertaken which allows for simultaneous estimation of the copula and the marginal distributions. Estimators based on various priors are analyzed via Markov chain Monte Carlo methodology. A rather extensive simulation experiment is carried out using data generated from parametric families of copulas. In many cases, the results indicate that the estimators obtained from the Bayesian approach outperforms the standard kernel approach in terms of mean integrated squared error.;Keywords. Bayes, Birkhoff polytope, Bivariate extreme value distribution, copulas, dependence function, doubly stochastic matrices, Gibbs, MCMC, Metropolis-Hastings, nonparametric, predictions, reversible jumps.