I wrote a blogpost on approximating the density of general sums of random variables with focus on negative binomials. Validated in Stan using SBC. Long story short: simple approximation by matching moments works very well. Still I hope there is something to be learned and the saddlepoint method is IMHO cool.
Hi, I just wanted to say that this post is great!
In my case (Using integration to fit the difference of gamma distributed random variable) , I actually found the saddle point analytically (well wolfram alpha did), and then it obviously goes super fast :), and it was a much more stable than the real approximate solution by integration.
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