General classes of bivariate distributions for modeling data with common observations

Abstract

In analyzing bivariate data sets, data with common observations are frequently encountered and, in this case, existing absolutely continuous bivariate distributions are not applicable. Only a few models, such as the bivariate distribution proposed by Marshall and Olkin (J Am Stat Assoc 62(317):30–44, 1967), have been developed to model such data sets and the choice of models to fit data sets having common observations is very limited. In this paper, three general classes of bivariate distributions for modeling data with common observations are developed. To develop the bivariate distributions, we employ a probability model in reliability. Considering a system with two components, it is assumed that, when the first failure of the components occurs, with some probability, it immediately causes the failure of the remaining component, and, with complementary probability, the residual lifetime of the remaining component is shortened according to some stochastic order. It will be shown that, by specifying the underlying distributions contained in the joint distribution, numerous families of bivariate distributions can be generated. Therefore, this work provides substantially increased flexibility in modeling data sets with common observations. The developed models are fitted to two real-life data sets and it is shown that these models outperform the existing models in terms of fitting performance and their performances are satisfactory.