DeepAI AI Chat
Log In Sign Up

Beta Rank Function: A Smooth Double-Pareto-Like Distribution

by   Oscar Fontanelli, et al.

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.


page 1

page 2

page 3

page 4


The nonlinear distribution of employment across municipalities

In this paper, the nonlinear distribution of employment across Spanish m...

Research assessment by percentile-based double rank analysis

In the double rank analysis of research publications, the local rank pos...

Scaling Properties of Human Brain Functional Networks

We investigate scaling properties of human brain functional networks in ...

Geometrically stopped Markovian random growth processes and Pareto tails

Many empirical studies document power law behavior in size distributions...

On Some Properties of the Beta Normal Distribution

The beta normal distribution is a generalization of both the normal dist...

Gauss and the identity function – a tale of characterizations of the normal distribution

The normal distribution is well-known for several results that it is the...

Two-Sided EWMA Charts for Monitoring Double Bounded Processes

In this work, we study the performance of two-sided EWMA charts for moni...