Parametric and non-parametric estimate of bivariate survival functions: the copula approach

In this paper we discuss the problem on parametric and non parametric estimation of the distributions generated by the Marshall-Olkin copula. This copula comes from the Marshall-Olkin bivariate exponential distribution used in reliability analysis. Through this copula we can extend the Marshall-Olkin distribution in order to construct several bivariate survival functions. The cumulative distribution functions of these distributions are not absolute continuous functions and they unknown parameters are often not be obtained in explicit form. In particular we consider the IFM method to find the Marshall-Olkin copula estimator, presenting the copula likelihood function. We compare this procedure with a non parametric estimator of the copula, the bivariate empirical copula, used to evaluate the copula goodness of fit. The estimate procedures described are verified through several simulation. One data-set is analyzed for a illustrative purpose.