I'm trying to compute some p-values for samples from a distribution of sums of ~1000 random variables. The exact distribution of these random variables isn't known, but I have empirical estimates that I think are pretty accurate.
So far I've been using the central limit theorem to produce a normal approximation for this sum, which does OK since n is relatively large, but not great. Computing the exact distribution of the sum via convolutions is too slow, but I only really care about having high accuracy in the tails; everywhere else a rough approximation is fine.
Are there any methods that will allow me to improve my estimates in the tails (or just one tail) without having to compute the entire convolutions? I'm not sure if this would be a variant on the CLT implementation or something completely different. My gut feeling is that there isn't any way to do this, so I'd be open to any kind of solution at all!