From Pareto to Weibull -- a constructive review of distributions on $\mathbb{R}^+$

TL;DR

This paper introduces the Interpolating Family of distributions on the positive real line, unifying power laws and Weibull distributions, and providing a comprehensive analysis of their properties and connections.

Contribution

It presents a novel family of distributions that bridges power laws and Weibull, offering new insights and encompassing various known and new distributions on \\mathbb{R}^+.

Findings

Unified treatment of power laws and Weibull distributions.

Derived properties like moments, quantiles, and modes for the new family.

Connected previously unrelated distributions through the Interpolating Family.

Abstract

Power laws and power laws with exponential cut-off are two distinct families of distributions on the positive real half-line. In the present paper, we propose a unified treatment of both families by building a family of distributions that interpolates between them, which we call Interpolating Family (IF) of distributions. Our original construction, which relies on techniques from statistical physics, provides a connection for hitherto unrelated distributions like the Pareto and Weibull distributions, and sheds new light on them. The IF also contains several distributions that are neither of power law nor of power law with exponential cut-off type. We calculate quantile-based properties, moments and modes for the IF. This allows us to review known properties of famous distributions on $\mathbb{R}^+$ and to provide in a single sweep these characteristics for various less known (and new) special cases of our Interpolating Family.