Stretched exponential distributions in Nature and Economy: ``Fat tails'' with characteristic scales

TL;DR

This paper introduces the stretched exponential distribution as a versatile model for fat-tailed data in Nature and Economy, highlighting its simplicity, physical interpretability, and broad applicability across various empirical datasets.

Contribution

It proposes the stretched exponential family as an effective alternative to power laws, with clear physical meaning and minimal parameters, supported by extensive empirical validation.

Findings

Stretched exponentials fit galaxy emissions, oil reserves, and city sizes well.

They describe Forex price variations and temperature fluctuations accurately.

Comparison shows advantages over other distributions like log-normal and fractals.

Abstract

To account quantitatively for many reported ``natural'' fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok temperature variations, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also briefly discuss its potential for the distribution of earthquake sizes and fault displacements and earth temperature variations over the last 400 000 years. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions.