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4

This is pretty tricky due to the very heavy tail of the distribution. You can certainly get big speed ups as suggested by bill s by pre-computing some quantiles. However, there will always be a good chunk of the tail left to compute. I'm going to try to address the latter and borrow from Bill's solution for pre-computation. s[n_] := Log[n]*n^(-1.5); A ...

5

If you are willing to precompute some things, it can be pretty quick. Here we precalculate 100000 terms of the $a_n$ sequence. Then calculate the CDF (cumulative distribution function) by using Accumulate. To find the closest term to the u, use a NearestFunction. capA = -Zeta'[3/2] // N; aAll = (Log[#]/#^1.5 & /@ Range[100000])/capA; accAll = ...

4

You specifically use the method of moments, is that meant to be a real problem constraint? If you can instead use the maximum likelihood estimate for {a,b} then you can you perform a straightforward likelihood ratio (LR) test. In the following quick check, the LR test statistic is approximately Chi-Square distributed (see Wilk's theorem), so you can accept ...

1

Because the distribution is clearly not a gamma distribution, finding estimates of gamma parameters won't be of much use. If there is a theoretical or historical reason to believe that a mixture of two or more distributions should describe the results of the process generating the data, then the above approaches are fine. But if you just need a description ...

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