New general classes of non-absolutely continuous bivariate distributions

In modeling bivariate data, frequently, the paired observations can coincide each other. In such cases, the existing absolutely continuous bivariate distributions cannot be applied. The most typical and popular bivariate model to analyze such bivariate data sets is the Marshall–Olkin Model (J Am Stat Assoc 62(317):30–44, 1967). Although some generalizations of the Marshall–Olkin Model have been proposed, the choice of models to fit bivariate data sets having tied observations is still limited. In this paper, general classes of non-absolutely continuous bivariate distributions for modeling bivariate data set having tied observations are developed. By employing a common shock model in reliability and combining it with existing absolutely continuous bivariate distributions, we develop three general classes of non-absolutely continuous bivariate distributions. It will be shown that the proposed classes of bivariate distributions are very general and flexible in the sense that, by specifying the underlying distributions contained in the joint distribution and the intensity function of the common shock process, numerous families of bivariate distributions can be generated. From the developed classes, several specific families of bivariate distributions are generated and their modeling performances are compared.