Convex Hull for Feasible Coefficients of ThreeTerm 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 FourTerm 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 limits, calculated via linear program, are shown in the table, where
 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 fourterm metalog is symmetric and peaked, similar to a studentt distribution with 3 degrees of freedom.
 At the bottom of this table, the fourterm 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 fourterm metalog is symmetric.
Below we illustrate the fourterm convex hull defined by the points in the table. The shaded area is feasible, while outside it is not.
Much of the feasible area can be closely approximated by an ellipse (dashed, gray curve), defined by center ${displaystyle b=4.5}$ and semiaxis lengths ${displaystyle c=8.5}$ and ${displaystyle d=1.95}$. Supplementing this with linear interpolation outside its applicable range, feasibility, given ${displaystyle a_{2}>0}$, can be closely approximated:


4term metalog feasibility limits (click for table in Excel) 

