A Method of moments to estimate bivariate survival functions: the copula approach

In this paper we study bivariate distributions used in reliability analysis by the meaning of copula. The copula is an instrument to generate bivariate and multivariate distributions (Nelsen, 2006 and Fisher, 1997). In particular we consider the survival copula of Marshall and Olkin. This copula comes from the bivariate Marshall-Olkin exponential distribution (Marshall and Olkin, 1967), proposed to study complex systems in which the two components are not independent. We generalize this model by the copula and different marginal distributions to construct several bivariate survival functions, i.e. bivariate Weibull distribution. These cumulative distribution functions are not absolutely continuous and their unknown parameters can not be obtained in explicit form by the maximum likelihood method. In order to estimate the parameters we propose an easy procedure based on the moments. This method consists 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 complete or censored sample (II Type) in order to evaluate the performance of the proposed estimation procedure