Generalized mixtures of exponential random variables (Everitt and Hand (1981) and Johnson et al. (1994)) X1 and X2 are identified in terms of their relations between the best predictor of X2:2 given X1:2 and the functions of the failure rate (or hazard function) of the distribution. Here X1:2 and X2:2 denote the corresponding order statistics. In addition, we also mention some related theorems in order to characterize the generalized mixtures of exponential distribution. Moreover, when the sample size is n, the above results are also valid.
Journal of the Japan Statistical Society 28(1), pp.39-44