I'm a bit of a stats newbie so take it easy on me if this ends up being somehow trivial. I'm working on a problem that involves parameter estimation for the sum of two independent gamma distributions (which do not necessarily have the same scale or shape parameters.) I'm interested in what would be the most convenient way to numerically estimate the parameters of the model. I'm primarily interested in approximating MLEs (though I would be interested in hearing alternative methods.) The convolution expression for the density seems a bit messy, and while I found a paper giving an expression for the density in terms of an infinite series with a nice truncation error estimate, the objective function is still pretty messy. Particularly, I'm curious to know where to start looking in terms of a) asymptotic approximation for the MLE and b) software that is well-suited for this type of application. Any links or references that you find useful will be most appreciated. Thanks!