Interesting question. It appears that the more general question (can you invert the function that calculates moments from a measure?) is called the "
Moment Problem". There are all sorts of studies into whether this inversion exists, and if it does, whether it is unique. I think in your case, you can afford to simplify the problem with the goal of actually calculating it at the end of the day.
I'm going to make a number of simplifications that may or may not apply in your scenario, but should help you make your own simplifications:
1. The random variable is discrete. The fact that you're talking about a histogram might satisfy this one nicely.
2. The range is finite. In fact, let's assume there are just 5 samples from 0 to 4. You can always scale appropriately.
3. The moments have all been calculated according to: u_n = integral( (x-c)^n * f(x), dx) which in the discrete case is re-written as a summation: u_n = sum(from 0, to 4, (x-c)^n * f(x)), where u_n is the nth moment, c is some central value, f(x) is your original (unknown) function and x is the sample. Often different order moments have slightly different calculating functions to normalise them or otherwise make them more useful. It shouldn't be a problem to revert them all to the same generating function.
4. c in the previous function is always 2. If it's not, that's no problem, it just has to be substituted appropriately.
With these assumptions, the derivation follows quite simply as a set of simultaneous equations:
u_0 = f0 + f1 + f2 + f3 + f4 (where the notation fn is used instead of f
for brevity)
u_1 = -2f0 - f1 + f3 + 2f4
u_2 = 4f0 + f1 + f3 + 4f4
u_3 = -8f0 - f1 + f3 + 8f4
Which are all linear, so you get all the normal results: there is a unique solution is you have as many moments as samples; and there are infinite solutions if you have less moments than samples.
The latter case is probably the more likely, so maybe it's worth pointing out that you can constrain your solution space quite dramatically if you know anything about the underlying distribution. If it is normal for example, just two moments (the mean and the standard deviation) is enough to uniquely determine the distribution.
Finally, I'd concur with GregLocock's excellent suggestion - since you're just after an approximation, a Monte Carlo type approach is well worth considering, especially if you only have a few moments and several samples to recover. As well as just minimising the error in the moments, you could help steer the Monte Carlo process with other soft information you have about the distribution. For example, you could start the Monte Carlo from a few distributions which are "close" to what you expect the final result to be and see if you can get them to converge.
