Some Problems on the Estimation of Marshall-Olkin Copula Parameters

This job discusses the problem of the study of the bivariate distributions used in reliability analysis by the meaning of a copula. The copula (Nelsen, 2006) is an helpful tool for handling multivariate distributions with given univariate marginals. For multivariate distribution, the univariate marginals and multivariate dependence structure can be separated, and the dependence structure can be represented by a copula. Copula allows to construct new multivariate distributions with arbitrary marginals. In this paper we consider the survival copula by Marshall and Olkin. This copula became by Marshall-Olkin bivariate exponential distribution (Marshall, Olkin, 1967). This model has been proposed to study complex system in which the two components are not independent. We extend this model using the copula and different marginal distribution in order to construct bivariate survival functions. The aim of this paper is to evaluate the problem of estimating the parameters of these distributions. We propose an easy procedure based on the method of moments, an alternative procedure to the maximum likelihood estimation generally used in literature. The moment’s method is used to estimate the distribution parameters in two steps: in the first step we estimate only the parameters of marginal distributions and in the second step we estimate only the copula parameter. The study of simulation is made either for the case of a complete or censored sample, this is done in order to evaluate the properties of the estimators for both cases.