The Difference-of-Log-Normals Distribution: Properties, Estimation, and Growth

Abstract

This paper describes the Difference-of-Log-Normals (DLN) distribution. A companion paper Parham (2023) makes the case that the DLN is a fundamental distribution in nature, and shows how a simple application of the CLT gives rise to the DLN in many disparate phenomena. Here, I characterize its PDF, CDF, moments, and parameter estimators; generalize it to N-dimensions using spherical distribution theory; describe methods to deal with its signature "double-exponential" nature; and use it to generalize growth measurement to possibly-negative variates distributing DLN. I also conduct Monte-Carlo experiments to establish some properties of the estimators and measures described.

Keywords: Heavy-tails, distributions, log-Normal, growth.

JEL Classification: C13, C46, C65