A Generalization of the Power Law Distribution with Nonlinear Exponent

TL;DR

This paper introduces the Generalized Power Law (GPL) distribution, a flexible model that accurately fits data across the entire range, including tails, by modeling the power law exponent as a nonlinear function.

Contribution

The paper proposes a new family of distributions, the GPL, with a nonlinear exponent, providing better modeling of data across all ranges compared to traditional power law models.

Findings

GPL fits municipal debt data across the entire range.

Power law is only valid in the upper tail of debt distribution.

GPL outperforms other distributions like Lognormal and Pareto in empirical tests.

Abstract

The power law distribution is usually used to fit data in the upper tail of the distribution. However, commonly it is not valid to model data in all the range. In this paper, we present a new family of distributions, the so-called Generalized Power Law (GPL), which can be useful for modeling data in all the range and possess power law tails. To do that, we model the exponent of the power law using a nonlinear function which depends on data and two parameters. Then, we provide some basic properties and some specific models of that new family of distributions. After that, we study a relevant model of the family, with special emphasis on the quantile and hazard functions, and the corresponding estimation and testing methods. Finally, as an empirical evidence, we study how the debt is distributed across municipalities in Spain. We check that power law model is only valid in the upper tail; we show analytically and graphically the competence of the new model with municipal debt data in the whole range; and we compare the new distribution with other well-known distributions including the Lognormal, the Generalized Pareto, the Fisk, the Burr type XII and the Dagum models.