Beta Rank Function: A Smooth Double-Pareto-Like Distribution
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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 from social science to biological phenomena. The underlying probability density function (pdf) f_X(x) does not usually have a closed 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 BRF pdf has simple properties when the independent variable is log-transformed: f_Z=log(X)(z) . At the peak it makes a smooth turn and it does not diverge, lacking the sharp angle observed in the double Pareto or Laplace distribution. The peak position of f_Z(z) is z_0=log A+(a-b)log(√(a)+√(b))-(alog(a)-blog(b))/2; the probability is partitioned by the peak to the proportion of √(b)/(√(a)+√(b)) (left) and √(a)/(√(a)+√(b)) (right); the functional form near the peak is controlled by the cubic term in the Taylor expansion when a b; the mean of Z is E[Z]=log A+a-b; the decay on left and right sides of the peak is approximately exponential with forms e^z-log A/b/b and e^ -z-log A/a/a. These results are confirmed by numerical simulations. Properties of f_X(x) without log-transforming the variable are much more complex, though the approximate double Pareto behavior, (x/A)^1/b/(bx) (for x<A) and (x/A)^-1/a/(ax) (for x > A) is simple. Our results elucidate the relationship between BRF and log-normal distributions when a=b and explain why the BRF is ubiquitous and versatile. Based on the pdf, we suggest a quick way to elucidate if a real data set follows a one-sided power-law, a log-normal, a two-sided power-law or a BRF. We illustrate our results with two examples: urban populations and financial returns.
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