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Pearson Distribution moment derivation

Posted 10 years ago
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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 ( 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]

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.
POSTED BY: Robert Hammond

Sorry for answering this so late ... I have only just seen the question (by chance). The Pearson family is conventionally defined in terms of 4 parameters (not 5). Solving the Pearson coefficients in terms of moments involves assuming the density tends to zero at the extremum of the tails, solving a recurrence relation, and is a little bit of work, but Mathematica does it very nicely indeed. See section 5.2D of our book, "Mathematical Statistics with Mathematica" which provides the derivation; if interested, a free download is available here:

Section 5.2E of Chapter 5 extends to a 5-parameter model based on a third-order (cubic) polynomial, instead of the standard second-order (quadratic) model ... and the same method works just as nicely.

POSTED BY: Crumpet Rasputin
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