How do you derive (or where can I find a derivation of) the equations to match the five parameters of a general Pearson distribution to a mean, variance, skewness, kurtosis, and condition that the function integrate to one?
I have been using the method described in the Mathematica documentation (http://reference.wolfram.com/mathematica/ref/PearsonDistribution.html) under "Examples" and then "Applications". It provides the equations but no derivation.
The four equations for the moments are set up as:
eq[r_] :=
r*Subscript[b, 0]*Moment[r - 1] + (r + 1) Subscript[b, 1]*Moment[r] + (r + 2) Subscript[b, 2]*Moment[r + 1] - Subscript[a, 1]*Moment[r + 1] - Subscript[a, 0]*Moment[r]
and
meq = Table[MomentConvert[eq[r], CentralMoment], {r, 0, 3}] /. {Moment[1] -> \[Mu], CentralMoment[2] -> \[Sigma]^2, CentralMoment[3] -> Sqrt[Subscript[\[Beta], 1]] \[Sigma]^3, CentralMoment[4] -> Subscript[\[Beta], 2] \[Sigma]^4}
These set up four linear equations in five unknowns. I am interested in the equation added to fix the normalizing coefficient (so the distribution integrates to one):
meq = Join[meq, {Subscript[a, 0] + (12 \[Mu] Subscript[\, 1] + 2 \[Mu] (9 - 5 Subscript[\, 2]) + \ Sqrt[Subscript[\, 1]] (3 + Subscript[\, 2]))}];
The equations for the moments themselves are pretty easy to derive, but I have not figured out the fifth equation for normalization.