The Metalog Distributions and Extremely Accurate Sums of Lognormals in Closed Form

Abstract

The metalog probability distributions can represent virtually any continuous shape with a single family of equations, making them far more flexible for representing data than the Pearson and other distributions. Moreover, the metalogs are easy to parameterize with data without non-linear parameter estimation, have simple closed-form equations, and offer a choice of boundedness. Their closed-form quantile functions (F <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ) enable fast and convenient simulation. The previously unsolved problem of a closed-form analytical expression for the sum of lognormals is one application. Uses include simulating total impact of an uncertain number N of risk events (each with iid [independent, identically distributed] individual lognormal impact), noise in wireless communications networks and many others. Beyond sums of lognormals, the approach may be directly applied to represent and subsequently simulate sums of iid variables from virtually any continuous distribution, and, more broadly, to products, extreme values, or other many-to-one change of iid or correlated variables.