Since the metalog quantile function is linear in its coefficients, it is natural to determine these coefficients by linear least squares (multiple linear regression). This method does not guarantee feasibility, which must be checked separately.
Linear Least Squares
Nine CDF points (orange) from various gamma distributions parameterize semi-bounded metalogs (dashed yellow). The metalog CDFs run exactly through each point. The metalog curves are virtually indistinguishable from the gamma curves.
Other Methods
While the above equations work well for most applications, the coefficients may be determined by other methods. For example, a least-absolute-distance (LAD) criterion with feasibility constraints may be implemented as a linear program. In the metalog packages in R and Python (rMetalog and pyMetalog available on GitHub), this LAD method is a backup in case the least-squares coefficients are infeasible.
While non-linear iterative optimization, such as least-squares with feasibility constraints, would rarely be useful, convex optimization could be used since the set of feasible metalog coefficients has been proven to be convex.
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