Reliability properties of order statistics from bivariate exponential distributions

Abstract

We explore the reliability properties of the minimum and maximum lifetimes of two–component systems whose components are dependent and have the bi–variate exponential (BVE) distributions of Gumbel, Marshall and Olkin, Block and Basu, Freund, Friday and Patil, and Raftery. In most cases, these random variables are either exponential or generalized hyperexponential (GH) distributions with three or fewer components. We determine the properties of the failure rates of such GH distributions in terms of the weights and the parameters of the constituent exponential random variables. We apply these general results and show that the failure rates of the order statistics from the above BVE distributions exhibit a variety of patterns. Their failure rate properties can differ substantially from those of order statistics of two independent exponential random variables