The Metalog Distributions

Convex Hull for Feasible Coefficients of Three-Term Metalogs

Feasibility condition for metalogs with ${displaystyle k=3}$ terms: ${displaystyle a_{1}}$ is any real number, ${displaystyle a_{2}>0}$ and ${displaystyle |a_{3}|/a_{2}leq 1.66711}$.

Convex Hull for Feasible Coefficients of Four-Term Metalogs

Feasibility for metalogs with ${displaystyle k=4}$ terms is defined as follows:

• ${displaystyle a_{1}}$ is any real number, and
• ${displaystyle a_{2}geq 0}$, and
• If ${displaystyle a_{2}=0}$, then ${displaystyle a_{3}=0}$ and ${displaystyle a_{4}>0}$ (uniform distribution exactly)
• If ${displaystyle a_{2}>0}$, then feasibility conditions are specified numerically
  • For a given ${displaystyle |a_{3}|/a_{2}}$, feasibility requires that ${displaystyle a_{4}/a_{2}geq }$ number shown.
  • For a given ${displaystyle a_{4}/a_{2}}$, feasibility requires that ${displaystyle |a_{3}|/a_{2}leq }$ number shown.
  • At the top of this table, the four-term metalog is symmetric and peaked, similar to a student-t distribution with 3 degrees of freedom.
  • At the bottom of this table, the four-term metalog is a uniform distribution exactly.
  • In between, it has varying degrees of skewness depending on ${displaystyle a_{3}}$. Positive ${displaystyle a_{3}}$ yields right skew. Negative ${displaystyle a_{3}}$ yields left skew. When ${displaystyle a_{3}=0}$, the four-term metalog is symmetric.

Below we illustrate the four-term convex hull defined by the points in the table. The shaded area is feasible, while outside it is not.

The feasible area can be closely approximated by an ellipse (dashed, gray curve), defined by center ${displaystyle b=4.5}$ and semi-axis lengths ${displaystyle c=8.5}$ and ${displaystyle d=1.93}$. Supplementing this with linear interpolation outside its applicable range, feasibility, given ${displaystyle a_{2}>0}$, can be closely approximated:

${\displaystyle \ \approx \left\{{\begin{array}{rlcrll}{|a_{3}| \over a_{2}}&\leq {d \over c}{\sqrt {c^{2}-({a_{4} \over a_{2}}-b)^{2}}}&{\text{ for }}&-4.0&\leq {a_{4} \over a_{2}}\leq 4.5,\\{|a_{3}| \over a_{2}}&\leq 0.0216({a_{4} \over a_{2}}-4.5)+1.930&{\mbox{ for }}&4.5&<{a_{4} \over a_{2}}\leq 7.0,\\{|a_{3}| \over a_{2}}&\leq 0.0040({a_{4} \over a_{2}}-7.0)+1.984&{\mbox{ for}}&7.0&<{a_{4} \over a_{2}}\leq 10.0,\\{|a_{3}| \over a_{2}}&\leq 0.0002({a_{4} \over a_{2}}-10.0)+1.996&{\mbox{ for}}&10.0&<{a_{4} \over a_{2}}\leq 30.0,\\{|a_{3}| \over a_{2}}&\leq 2.0&{\mbox{ for}}&30.0&<{a_{4} \over a_{2}}.\end{array}}\right.}$