Reverse engineer a new set of points from an original set by altering moments, skew, and/or Kurtosis?

Question

I don't know if this is even possible but I'd like to be able to take a set of points, run something on them that calculates the moments, skew and kurtosis values, and have another function that would take those elements and reverse engineer a new set of points using modified values for the moments, skew and/or kurtosis. I already have the analytical function in Delphi Pro 6 which is:

procedure MomentSkewKurtosis(const Data: array of Double;var M1, M2, M3, M4, Skew,Kurtosis: Extended);

I'm looking for a partner function that could return a new Data array after I make alterations to any of the output parameters "var" in MomentSkewKurtosis() and pass them back in to the partner function as input parameters. For example, suppose I wanted to increase the Skew of the data and get a new set of points back that would be the original set of points altered just enough to generate the new Skew value.

Answer 1

The problem is not easy, and probably better targetted at stats, but I'll give you a pointer to a paper that I think is very good, and straight to the mark: Towards the Optimal Reconstruction of a Distribution from its Moments

Hope this helps!

Answer 2

Obviously you can't reconstruct an arbitrary density distribution from a finite amount of variables. You can create a distribution which fits the parameters, but it's not necessarily the original distribution.

And as far as I remember Mean, Variance, Skew and Kurtosis are just functions of the first 4 momenta. So you can't choose them independently from the momenta.

On the other hand there exists a function which you can apply on each data member and that produces a new dataset with the desired properties. I suspect that since you fixed the first 4 momenta it's a polynomial of grade 3.