Survival analysis of a new compounded bivariate failure time distribution and its generalization

In this paper, we consider that two bivariate models based on the Gompertz distribution and the proposed methods of Marshall and Olkin (1967) in bivariate case and Marshall and Olkin (1997) in the univariate cases. In the second case, their method is generalized to bivariate case and a new bivariate distribution is introduced. These new bivariate distributions have natural interpretations, and they can be applied in fatal shock models or in competing risks models. We call these new distributions as the Marshall Olkin bivariate Gompertz (MOBGP) distribution and bivariate Gompertz-geometric (BGPG) distribution, respectively. Moreover, the MOBGP can be obtained as a special case of the BGPG model. Then, the various properties of the new distributions are investigated. The BGPG distribution has five parameters and the maximum likelihood estimators cannot be obtained in closed form. We propose to use the EM algorithm, and it is observed that the implementation of the EM algorithm is quite straightforward. Also, one data analysis is provided for illustrative purposes.