“The Difference-of-Log-Normals Distribution is Fundamental in Nature” [PDF]
The growth of many natural and social phenomena including pandemics, firms, cities, and various economic indices, is known to be heavy-tailed. Most growth is modest, but we often observe explosive growth rates, such as firms doubling or halving in size within a short period. Neither a simple explanation nor a well-fitting distributional form for these growth phenomena is known. Here I show that a hitherto obscure statistical distribution — the Difference-of-Log-Normals (DLN) — describes a plethora of growth phenomena remarkably well, and discuss why it arises as a natural consequence of the Central Limit Theorem (CLT). The results demonstrate how growth phenomena subject to opposing random exponential forces are likely to distribute DLN. This provides both a framework for scientifically modeling these phenomena and a simple distributional form to be used when empirically modeling observed heavy-tailed growth. I hence posit that the DLN is a fundamental distribution in nature, in the sense that it emerges in many disparate natural phenomena, especially growth phenomena, similar to the repeated disparate emergence of the Normal and log-Normal distributions.
“The Difference-of-Log-Normals Distribution: Properties, Estimation, and Growth” [arXiv]
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.
“Facts of US Firm Scale and Growth 1970-2019: An Illustrated Guide” [arXiv]
This work analyzes data on all public US firms in the 50 year period 1970-2019, and presents 18 stylized facts of their scale, income, growth, return, investment, and dynamism. Special attention is given to (i) identifying distributional forms; and (ii) scale effects — systematic difference between firms based on their scale of operations. Notable findings are that the Difference-of-Log-Normals (DLN) distribution has a central role in describing firm data, scale-dependent heteroskedasticity is rampant, and small firms are systematically different from large firms.