Arbitrary nonparametric mixtures of exponential and Weibull (fixed shape) distributions are considered as possible models for a lifetime distribution. A characterization of such distributions is given by the well-known characterization of Laplace transforms. The maximum likelihood estimate of the mixing distribution is investigated and found to be supported on a finite number of points. It is shown to be unique and weakly convergent to the true mixing measure with probability one. A practical algorithm for computing the maximum likelihood estimate is described. Its performance is briefly discussed and some illustrative examples given.
"Mixtures of Exponential Distributions." Ann. Statist. 10 (2) 479 - 484, June, 1982. https://doi.org/10.1214/aos/1176345789