User: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
 * $$a_2\geq0$$, 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. $$