Estimation de paramètres en exploitant les aspects calculatoires et numériques

In this thesis, we are mainly interested in the estimation of parameters. It consists of three papers in which we discuss estimation problems in specific models.  In the first paper, we consider the Farlie-Gumbel-Morgenstern parametric family of copulas. The observations come from a distribution that involves both the copula and the margins, through the decomposition of the Sklar's theorem. Margins are unknown. In the estimation of a function of the parameter of the copula, pseudo-likelihood is often used instead of the likelihood. It is an approach that will generally produce non-efficient estimators, especially for small sample sizes. In pseudo-likelihood, margins are replaced by estimators that depend on the ranks. We propose to use the rank-likelihood, which is the likelihood based on the distribution of the rank statistics. This approach can be complex in practice because we work on the permutation group and we calculate multiple integrals. However, it is possible to do the calculations for the Farlie-Gumbel-Morgenstern family of copulas. We compare the estimators obtained from these approaches using the Bayesian method. The results of numerical studies are presented.    Some estimators found in the literature are ratios of normal random variables. In the second paper, we are interested in the distribution of the ratio of two normal random variables. Several authors have addressed this subject. The new parametrization that we introduced allows us to obtain new results faster and effortlessly. Firstly, we show that the expression of the density of the ratio can be written as a mixture of densities belonging to a new family that we create. We find some properties, and we give the analytic expression of the characteristic function for the new family. This new family is in fact a generalization of the family of densities of Student distributions having odd degrees of freedom. We obtain convergence results  generalizing the one for Student distributions when the degree of freedom tends to infinity. Similar results of convergence as well as some properties are developed for the distribution of the ratio. We use a Bayesian approach to estimate the ratio of the means of two normal random variables. Graphical illustrations and simulation results are presented.   In the third paper, we extend the family of distributions, resulting from the difference of two independent random variables having a gamma distribution, by considering the case where these random variables are positively correlated. The densities in this family are found. Their complex forms make tedious the estimation methods based on the likelihood. The estimation method based on the characteristic function is used, and the continuous approach is compared to the discrete approach. Moreover, we developed two simple algorithms for generating data from two positively correlated and identically distributed random variables having a gamma distribution. The best equivariant estimator for the scale parameter is obtained in the case of independence.

Keyphrases: Bayesian method, Characteristic function, Convergence of random variables, Copula, Difference of gamma variates, Equivariant estimator, Parameter estimation, Pseudo-likelihood, Rank-likelihood, Ratio of normal variables