The Metalog Distributions

Feasibility

For a given set of coefficients, a metalog is a feasible probability distribution if and only if its PDF m(y)>0 for all 0<y<1. Generally, this condition is checked numerically or graphically. However, for 2-4 term metalogs, the feasibility condition can be written in closed form. For 2 terms, feasibility requires that a2>0. For 3 and 4 terms, this condition is given below.


For a given number of terms, all members of the metalog family (including semibounded and bounded metalogs) share the same set of feasible coefficients. Thus, the conditions below apply equally to all of them.


Convex Hull for Feasible Coefficients of Three-Term Metalogs


Feasibility condition for metalogs with terms: is any real number, and .


Convex Hull for Feasible Coefficients of Four-Term Metalogs


Feasibility for metalogs with terms is defined as follows:

  • is any real number, and
  • , and
  • If , then and (uniform distribution exactly)
  • If , then feasibility limits, calculated via linear program, are shown in the table, where
    • For a given , feasibility requires that number shown.
    • For a given , feasibility requires that 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 . Positive yields right skew. Negative yields left skew. When , 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.


Much of the feasible area can be closely approximated by an ellipse (dashed, gray curve), defined by center and semi-axis lengths and . Supplementing this with linear interpolation outside its applicable range, feasibility, given , can be closely approximated:


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