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

TL;DR

This paper thoroughly characterizes the Difference-of-Log-Normals distribution, exploring its properties, estimation methods, and applications to growth measurement, supported by Monte Carlo experiments.

Contribution

It introduces a comprehensive analysis of the DLN distribution, including its properties, multidimensional generalization, and novel methods for growth measurement.

Findings

Characterized the PDF, CDF, and moments of DLN

Developed parameter estimators and generalized to N-dimensions

Validated properties through Monte Carlo experiments

Abstract

This paper describes the Difference-of-Log-Normals (DLN) distribution. A companion paper 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.