Beta rank function: A smooth double-Pareto-like distribution
Por:
Fontanelli, Oscar, Miramontes, Pedro, Mansilla, Ricardo, Cocho, Germinal, Li, Wentian
Publicada:
3 jun 2022
Ahead of Print:
1 ene 2020
Categoría:
Statistics and probability
Resumen:
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. The underlying probability
density function (pdf) f(X)(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 f(Z
= log X)(z) is smooth at the peak; probability is partitioned by the
peak with proportion root b/root a (left to right); decay on left and
right tails is approximately exponential, e(z/log (A)/b) / b and
e(-z-log(A)/a) / a respectively. On the other hand, f(X) (x) behaves
like a power distribution x(1/b-1) when x similar to 0 and decays like a
Pareto 1/x(1/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.
Filiaciones:
Fontanelli, Oscar:
Centro de Investigaciones Interdisciplinarias en Ciencias y Humanidades, Universidad Nacional Autónoma de Mexico, Mexico City, Mexico
Miramontes, Pedro:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de Mexico, Mexico City, Mexico
Mansilla, Ricardo:
Centro de Investigaciones Interdisciplinarias en Ciencias y Humanidades, Universidad Nacional Autónoma de Mexico, Mexico City, Mexico
Cocho, Germinal:
Instituto de Fisica, Universidad Nacional Autonoma de Mexico, Mexico City, Mexico
Li, Wentian:
The Robert S. Boas Center for Genomics and Human Genetics, The Feinstein Institutes for Medical Research, Manhasset, NY, United States
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