I have seen several models for estimating the expected value and variance of a distribution. I am curious, to learn if anyone has looked at models that extend beyond these first two moments especially skew & kurtosis. I will be grateful if you could share your thoughts.
The Pearson family and Johnson family are both based on the first 4 moments ... and, in particular, expressed in terms of skewness and kurtosis. Both approaches generally work very well provided skewness and kurtosis are not too large and your model is unimodal.
For application and much more detail, see, for instance:
Chapter 5 of our book: Rose and Smith, Mathematical Statistics with Mathematica $\rightarrow$ a free download is available at: http://www.mathstatica.com/book/bookcontents.html , or
Stuart and Ord (1994), Kendall's Advanced Theory of Statistics (6th edition) - Chapter 6.
For an example of Johnson fitting on stackexchange, see this question: Robust distribution fitting?