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

The beta rank function (BRF), x(u)=A(1−u)b/ua, 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) fX(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 fZ= log X(z) is smooth at the peak; probability is partitioned by the peak with proportion b/a (left to right); decay on left and right tails is approximately exponential, ez− log (A)b/b and e−z− log (A)a/a respectively. On the other hand, fX(x) behaves like a power distribution x1/b−1 when x∼0 and decays like a Pareto 1/x1/a+1 when x≫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.