The Metalog Distributions

Metalog Flexibility Theorem

Theorem: Any probability distribution with a continuous quantile function can be approximated arbitrarily closely by a metalog quantile function.

Proof: Let  be an arbitrarily small positive number and let  be a continuous quantile function defined on the probability interval . By the Weierstrass approximation theorem, for every there exists a polynomial  such that  for all .  By setting the to zero for all terms that include the factor , the metalog quantile function reduces to a polynomial.  Therefore, there exists a metalog distribution  such that  for all .

While this theorem guarantees the existence of a such a metalog for any continuous quantile function, it does not guarantee its feasibility. It does guarantee, however, that there exists a metalog that is everywhere within  of being feasible for any arbitrarily small positive number , which should be sufficient for most practical applications.

Nor does this theorem provide a procedure for how to find the metalog  coefficients that correspond to an arbitrary continuous quantile function. Nor does it guarantee rapid convergence as the number of terms increases. But in practice, metalogs do converge rapidly and the coefficients can easily be determined by least squares. For example, we showed in the original paper, Table 8, that metalog distributions parameterized by 105 CDF points from 30 traditional source distributions (including the normal, student-t, lognormal, gamma, beta, and extreme-value distributions) approximate each such source distribution within a K-S distance of 0.001 or less as the number of metalog terms approaches ten.

More broadly, this theorem equally applies to any quantile-parameterized distribution containing a polynomial with an unlimited number of terms.