Like a Taylor series, the metalog quantile function may have any number of terms. Each additional term defines a different probability distribution and adds shape flexibility. Here we explore how closely 2-10 term metalogs approximate traditional probability distributions and shape themselves to empirical data sets.
The original paper showed that metalogs parameterized by 105 cumulative distribution function (CDF)-data points from each of 30 traditional source distributions (including normal, lognormal, Weibull, gamma, chi-squared, beta, extreme value, and student-t) converged rapidly to those source distributions in terms of Kolmogorov-Smirnov criterion as the number of terms. Moreover, the source vs metalog PDFs for all 30 distributions become visually indistinguishable as the number of metalog terms approaches 10.
Below is an example. We used 105 points from CDF of a standard normal distribution to parameterize unbounded metalog distributions. These 105 points correspond to y-probabilities of (0.001, 0.003, 0.006, 0.01, 0.02, ... , 0.99, 0.994, 0.997, 0.999). As the number of terms increases from 2 to 10, the metalog PDF and CDF become visually indistinguishable from the source distribution.
2-10 term metalog distributions parameterized by 105 points from standard normal CDF
2-10 term metalog approximations of standard normal distribution