Estimation problems based on dependent data

Résumé

The work in this dissertation focuses on estimation problems for copula-based Markov chains, which best describe the dependence between data. Several issues are addressed in this dissertation, among which: the study of copula modification by the perturbation method, the study of dependence properties of copula-based Markov chains, the study of mixing properties of copula-based Markov chains and the study of parameter estimation procedures for copula-based models. Past studies on copula estimation in the literature are based on MLE (Maximum likelihood Estimation), IFM (Inference Function for Margins) and methods of moments. We tackle the problem using perturbations of copulas and the properties of linear operations generated by these copula forms. More specifically, we explore the impact of perturbations of copulas on the dependence properties of the Markov chains they generate. We provide results for the mixing coefficients $\beta_n$, $\psi_n$, $\phi_n$. Also, a valid observation is used for convex combinations of copulas to establish sufficient conditions for the mixing coefficients $\rho_n$, $\alpha_n$ and other measures of association. New copula functions are provided in connection with perturbations of variables that induce other types of copula perturbations not considered in the literature. In addition, new families of copulas are derived from the basis of copula perturbations and their multivariate analogues for $n$-copulas are provided in general. On the basis of the copulas obtained by the perturbation method, we estimate the parameters of a stationary Markov chain whose copula was developed by Longla \cite{Longla4}. We use Maximum Likelihood estimation method, two step pseudo-maximum likelihood estimation method and the method of Longla and Mous-Abou \cite{LonglaMous}.

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