Return the Gaussian expanded pdf function given the list of 1st, 2nd moment and skew and Fisher (excess) kurtosis.
distribution is matched to these four moments
function that evaluates the pdf(x), where x is the non-standardized random variable.
Changed so it works only if four arguments are given. Uses explicit formula, not loop.
This implements a Gram-Charlier expansion of the normal distribution where the first 2 moments coincide with those of the normal distribution but skew and kurtosis can deviate from it.
In the Gram-Charlier distribution it is possible that the density becomes negative. This is the case when the deviation from the normal distribution is too large.
https://en.wikipedia.org/wiki/Edgeworth_series Johnson N.L., S. Kotz, N. Balakrishnan: Continuous Univariate Distributions, Volume 1, 2nd ed., p.30