Jade Tiger noticed a very interesting result from taking the consecutive differences of consecutive powers. The proof can now be seen in the pictures just be sure to read the captions to make sure you are following the correct page order.
If you have any questions with this proof, please post them here.
There are a few different methods to do this with.
Although first you need to know what type of distribution you want to fit it with.
Then the methods I'm familiar with are percentile matching (try to match percentiles to the empirical data), maximum likelihood estimators (for lots of data this could be a bit cumbersome), or the method of matching moments.
In all these cases you'll need as many equations as the underlying distribution has parameters.
Now if you're a bit unsure about what distribution you can use the maximum likelihood method on a few different distributions and then there is a the Schwarz Bayesian Criterion to determine whether or not you get a significantly better fit by adding another parameter (ie. moving from exponential to gamma)
So the maximum likelihood ones are a bit tough to explain here, but I can explain the percentile matching and moment matching.
So pick your distribution with n parameters
Pick n percentiles (ie. 50th, 75th)
You now have n equations for the n percentiles and n unknowns.
(this may be really hard to solve depending on the distribution, solver in excel might help).
For method of moments calculate the first n moments of the distribution then you'll have n equations and n unknowns.
Again this might be tough, as well some distributions don't have a finite n'th moment.
How many data points are we talking about? And what kinds of distributions are you thinking? Maximum likelihood might be your best bet.
so it seems like the algorithm you have already uses some version of MLE. I'm not too familiar with the algorithms for this, as I've just used it by hand before, and depending on the distribution it's not too bad. But I'd just use the MLE with the Schwarz Bayesian Criterion to see if more parameters are warranted.
Or you could use the chi-square statistic and set up a formula in excel and use solver to minimize it.... again you'll need to have a sheet or something for each possible distribution and see which one has the best chi-square.