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 ∈ (0, 1] is the normalized and continuous rank of an observation x, has wide applications in fitting real-world data from social science to biological phenomena. The probability density function (pdf) converted from the BRF, f X (x), does not usually have an analytic expression except for specific parameter values. We show however that it is approximately a unimodal skewed and asymmetric two-sided power law/double Pareto/log-Laplacian distribution. The pdf of the BRF has very simple properties when the independent variable is log-transformed: f Z=log(X) (z) .