# The Metalog Distributions

 Unbounded Metalog Moments The moments of the unboounded metalog distribution can be all expressed as a closed-form function of the scaling constants (a1, ... an). Generally, the kth central moment of the n term metalog is a kth-order polynomial of the ai's.  See The Metalog Distributions, Section 3.4, for further explanation.  You may download these moments below.
 To view the first four central moments (mean, variance, skewness, and kurtosis) for unbounded metalog distributions of up to ten terms, click the icon on the right.  For the Excel implementation of these moments, download the "Unbounded Metalog" Excel workbook. First Four Moments For Metalogs Up to Ten Terms
 To view the first four central moments (mean, variance, skewness, and kurtosis) for unbounded metalog distributions of up to sixteen terms, click the icon on the right.  The equations provided are for the 16 term unbounded metalog. For moments of metalogs with fewer terms, set the higher-order a-coefficients to zero. For the Excel implementation of these moments, download the "Unbounded Metalog" Excel workbook. First Four Moments for Metalogs with up to 16 terms
 To view the first ten central moments for unbounded metalog distributions of up to five terms, click the icon on the right. First Ten Moments For Metalogs Up To Five Terms
 Parameterization with Moments Three- and four-term unbounded metalogs can be parameterized in closed form with moments.       Parameterizing Three-Term Metalog with First Three Central Moments
 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 ${\displaystyle {|a_{3}|/a_{2}}<1.66711}$. . . Click here for proof.
 Parameterizing Four-Term Metalog with First Four Central Moments (Equations and methods to be published here soon.)
 Semi-Bounded and Bounded-Metalog Moments The moments of semi-bounded and bounded metalog distributions are not available in closed form. They must be calculated by numerical integration, for example by quadrature.