Abstract
The beta rank function (BRF), 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)
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
is smooth at the peak; probability is partitioned by the peak with proportion
(left to right); decay on left and right tails is approximately exponential,
and
respectively. On the other hand,
behaves like a power distribution
when
and decays like a Pareto
when
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.
Acknowledgements
This paper is dedicated to the memory of Prof. Germinal Cocho, who passed away during the process of its elaboration.