Tomkeelin

Another generalized log-logistic distribution is the log-transform of the metalog distribution, in which power series expansions in terms of $p$ are substituted for logistic-distribution parameters $\mu$ and $\sigma$. The resulting metalog quantile function is highly shape flexible, has a simple closed form, and can be fit to data with linear least squares. The log-logistic distribution is special case of the log-metalog distribution.

Convex Hull for Feasible Coefficients of Three-Term Metalogs

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

Convex Hull for Feasible Coefficients of Four-Term Metalogs

Convex Hull for Feasible Coefficients of Four-Term Metalogs

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

• $a_{1}$ is any real number, and
• $a_{2}\geq 0$, and
• If $a_{2}=0$, then $a_{3}=0$ and $a_{4}>0$ (uniform distribution exactly)
• If $a_{2}>0$, then feasibility conditions are specified numerically
  • For a given $|a_{3}|/a_{2}$, feasibility requires that $a_{4}/a_{2}\geq$ number shown.
  • For a given $a_{4}/a_{2}$, feasibility requires that $|a_{3}|/a_{2}\leq$ 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 $a_{3}$. Positive $a_{3}$ yields right skew. Negative $a_{3}$ yields left skew. When $a_{3}=0$, the four-term metalog is symmetric.

Convex Hull Equations

The feasible area can be closely approximated by an ellipse (dashed, gray curve), defined by center $b=4.5$ and semi-axis lengths $c=8.5$ and $d=1.95$. Supplementing this with linear interpolation outside its applicable range, feasibility, given $a_{2}>0$, can be closely approximated:

$\ \approx \left\{{\begin{array}{rlcrll}{|a_{3}| \over a_{2}}&\leq {d \over c}{\sqrt {c^{2}-({a_{4} \over a_{2}}-b)^{2}}}&{\text{ for }}&-4.0&\leq {a_{4} \over a_{2}}\leq 4.5,\\{|a_{3}| \over a_{2}}&\leq 0.014({a_{4} \over a_{2}}-4.5)+1.95&{\mbox{ for }}&4.5&<{a_{4} \over a_{2}}\leq 7.0,\\{|a_{3}| \over a_{2}}&\leq 0.004({a_{4} \over a_{2}}-7.0)+1.984&{\mbox{ for}}&7.0&<{a_{4} \over a_{2}}\leq 10.0,\\{|a_{3}| \over a_{2}}&\leq 2.0&{\mbox{ for}}&10.0&<{a_{4} \over a_{2}}.\end{array}}\right.$