q-exponential, Weibull, and q-Weibull distributions: an empirical analysis

TL;DR

This paper compares q-exponential, Weibull, and introduces q-Weibull distributions to model various empirical data, revealing which distributions best fit different real-world phenomena and shedding light on the stretched exponential versus power law debate.

Contribution

It introduces the q-Weibull distribution as an interpolating model between q-exponential and Weibull, providing a better fit for complex data sets.

Findings

Basketball data fits q-exponential well.

Cyclone victims and drug sales fit Weibull.

Highway length data requires q-Weibull for proper fit.

Abstract

In a comparative study, the q-exponential and Weibull distributions are employed to investigate frequency distributions of basketball baskets, cyclone victims, brand-name drugs by retail sales, and highway length. In order to analyze the intermediate cases, a distribution, the q-Weibull one, which interpolates the q-exponential and Weibull ones, is introduced. It is verified that the basketball baskets distribution is well described by a q-exponential, whereas the cyclone victims and brand-name drugs by retail sales ones are better adjusted by a Weibull distribution. On the other hand, for highway length the q-exponential and Weibull distributions do not give satisfactory adjustment, being necessary to employ the q-Weibull distribution. Furthermore, the introduction of this interpolating distribution gives an illumination from the point of view of the stretched exponential against inverse power law (q-exponential with q > 1) controversy.