The Metalog Distributions

Let ${displaystyle m,v,}$ and ${displaystyle s}$ be the mean, variance, and skewness respectively, and let ${displaystyle s_{s}}$ be the standardized skewness, ${displaystyle s_{s}=s/v^{3/2}}$. Equivalent expressions of the moments in terms of the coefficients and coefficients in terms of the moments are as follows.

${\displaystyle {\begin{aligned}m={}&a_{1}+{a_{3} \over 2}\\[6pt]v={}&\pi ^{2}{{a_{2}}^{2} \over 3}+{{a_{3}}^{2} \over {12}}+\pi ^{2}{{a_{3}}^{2} \over {36}}\\[6pt]s={}&\pi ^{2}{a_{2}}^{2}{a_{3}}+\pi ^{2}{{a_{3}}^{3} \over {24}}\\[6pt]\end{aligned}}}$

${\displaystyle {\begin{array}{l}a_{1}=m-{a_{3} \over 2}\\a_{2}={1 \over {\pi }}{\Bigl [}3{\Bigl (}v-{\Bigl (}{1 \over {12}}+{{\pi }^{2} \over {36}}{\Bigr )}{a_{3}}^{2}{\Bigr )}{\Bigr ]}^{1 \over {2}}\\a_{3}=4{\Bigl (}{6v \over {6+\pi ^{2}}}{\Bigr )}^{1 \over {2}}\cos {\Bigl [}{1 \over {3}}{\Bigl (}\cos ^{-1}{\Bigl (}-{s_{s} \over {4}}{\Bigl (}1+{{\pi }^{2} \over {6}}{\Bigr )}^{1 \over {2}}{\Bigr )}+4\pi {\Bigr )}{\Bigr ]}\end{array}}}$

To validate this equivalence, start with a given set of moments and calculate the corresponding coefficients with the equations on the right. Then take these coefficients, fill them into the equations on the left, and note that the result is exactly the set of moments with which you started. We derived this result by noting that the equations on the left reduce to a cubic polynomial in terms of the coefficients, which can be solved in closed form in terms of the moments. Moreover, this solution is unique. In terms of moments, the feasibility condition is ${\displaystyle |s_{s}|\leq 2.07093}$, which can be shown to be equivalent to the feasibility condition in terms of coefficients: ${\displaystyle a_{2}>0}$ and ${\displaystyle {|a_{3}|/a_{2}}<1.66711}$.