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During the execution of a script I wrote, the answer is a stochastic variable with an unknown distribution.

While I do not know the distribution of the answer, I do have access to the following information:

  • Moments can be approximated pretty well with standard quadratures
  • Random values can be evaluated very fast
  • Most of the time I have a rough estimation of the support of the distribution

What would be the best approach (as in: best approximation given a fixed time bound) to finding an estimation for the probability distribution?

I could draw random values, and use the resulting histogram as an approximation, but this is slow converging. I have not yet tried entropy maximizing algorithms, but I suppose those could work as well.

Does anyone have a better suggestion?

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Consider kernel density estimation or orthogonal series density estimation (e.g., wavelet density estimation).

Do you know the support (non-zero domain) of the density?

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    yes, with 'carrier' I meant 'support', sorry (translation error)2011-11-19
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    As far as I see, 'kernel density estimation' is a fancy term for smoothing the curve? That's great for displaying the result, but it doesn't decrease the absolute error compared to a normal histogram. I was actually looking for a method to make better use of the moments (or at least a method to combine both the information of the random samples and the moments) (but thanks for the answer anyway ;-) )2011-11-19
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    No, that's not true. Another question then is are you dealing with a discrete distribution or a continuous distribution? I assumed your histogram was made by binning but now I suspect you have a discrete distribution.2011-11-19
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    No, the output is a continuous distribution, but the approximation with sample points is (obviously) discrete. In any way, a discrete approximation will do. I am not really looking for any nice looking smooth curves, I really just want to make better use of the information I have than just sampling the distribution (normalizing to the known moments seems to help a little)2011-11-19
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    Smoothing is not superfluous; it is done to avoid [overfitting](http://en.wikipedia.org/wiki/Overfitting).2011-11-19