Some statistical inferences for the bivariate exponential distribution

The bivariate exponential distribution is neither absolutely continuous nor discrete due to the property that there is a positive probability that the two random variables may be equal. Basic properties of the distribution are presented as well as methods of parameter estimation including maximum likelihood. The distribution is shown to satisfy the usual regularity conditions in spite of its possession of a singularity. The maximum likelihood estimates are asymptotically efficient. Two other methods of estimation are compared with the maximum likelihood method in terms of efficiency. Tests of the hypothesis that two random variables each have independent exponential distributions versus the alternative hypothesis that the variables follow a bivariate exponential distribution with positive correlation are considered in detail. The estimation of the reliability of a simple two component series system or a parallel system in which the components have life times which follow the bivariate exponential distribution is considered. The errors made when assuming erroneously that the two random variables are independent, each with exponential distributions, when in fact they follow the bivariate exponential distribution, are illustrated --Abstract, page ii