Beta rank function: A smooth double-Pareto-like distribution

Abstract

The beta rank function (BRF), $x(u)=A{(1-u)}^b/u^a,$ where u is the normalized and continuous rank of an observation x, has wide applications in fitting real-world data. The underlying probability density function (pdf) $f_X(x)$ is not expressible in terms of elementary functions except for specific parameter values. We show however that it is approximately a unimodal skewed two-sided power law, or double-Pareto, or log-Laplacian distribution. Analysis of the pdf is simplified when the independent variable is log-transformed; the pdf $f_{Z=\hspace{0.17em}\text{log}\hspace{0.17em}X}(z)$ is smooth at the peak; probability is partitioned by the peak with proportion $\sqrt{b}/\sqrt{a}$ (left to right); decay on left and right tails is approximately exponential, $e^{\frac{z-\hspace{0.17em}\text{log}\hspace{0.17em}(A)}{b}}/b$ and $e^{-\frac{z-\hspace{0.17em}\text{log}\hspace{0.17em}(A)}{a}}/a$ respectively. On the other hand, $f_X(x)$ behaves like a power distribution $x^{1/b-1}$ when $x\sim 0$ and decays like a Pareto $1/x^{1/a+1}$ when $x\gg 0.$ We give closed-form expressions of both pdf’s in terms of Fox-H functions and propose numerical algorithms to approximate them. We suggest a way to elucidate if a data set follows a one-sided power law, a lognormal, a two-sided power law or a BRF. Finally, we illustrate the usefulness of these distributions in data analysis through a few examples.

Keywords:

• Pareto distribution
• power law
• quantile function
• rank-order function
• discrete generalized beta distribution

Acknowledgements

This paper is dedicated to the memory of Prof. Germinal Cocho, who passed away during the process of its elaboration.

Additional information

Funding

PM acknowledges support from Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (PAPIIT IN108318). OF is a grant holder of the Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México - Postdoctoral Scholarships Program at CEIICH, UNAM, working under supervision of Ricardo Mansilla Corona and Aquiles Negrete Yankelevich. WL acknowledges the support from Robert S. Boas Center for Genomics and Human Genetics.