Generalized Mixtures of Gamma and Exponentials and Reliability Properties of the Maximum from Friday and Patil Bivariate Exponential Model

The minimum and maximum order statistics from many of the common bivariate exponential distributions are predominantly generalized mixtures of exponentials; however, the maximum from the Friday and Patil bivariate exponential (FPBVE) model is either a generalized mixture of three or fewer exponentials or a generalized mixture of gamma and exponentials. In this article, we obtain conditions based on the weights and parameters of the generalized mixtures of gamma and one or two exponential distributions that yield legitimate probability models. Furthermore, we analyze properties of the failure rate of the maximum from the FPBVE model. This answers a question raised in Baggs and Nagaraja (1996 Baggs , G. E. , Nagaraja , H. N. ( 1996 ). Reliability properties of order statistics from bivariate exponential distributions . Commun. Statist. Stochastic Mod. 12 : 611 – 631 .Taylor & Francis Online] , Google Scholar]).