# Parameter estimation for the sum of two Independent (not necessarily i.d.) Gamma RVs

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!

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It is easy to implement the density (and therefore the likelihood) in R using numerical integration, but a major concern here would be parameter identifiability. I tried to approximate the MLE in this way but I found (empirically) this issue. –  user10525 Jun 29 '12 at 12:10
Yeah, the possibility of numerical insensitivity made me hesitant to use any built-in integration function. –  Tim Duff Jun 29 '12 at 21:32
As Procrastinator said, you will face identifiability problems and need inequality constraints on the parameters. In the frequentist context, you can use the Expectation-Maximisation (EM) algorithm. This needs here to compute numerical densities and conditional expectations by using the Discrete Fourier Transform (DFT). It will require writing a program (R, octave or Matlab), and also testing it carefully. You can also use similar ideas in a bayesian framework. –  Yves Jul 1 '12 at 12:47