Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Beta Rank Function: A Smooth Double-Pareto-Like Distribution   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(\sqrt{a}+\sqrt{b})-(a\log(a)-b\log(b))/2 $; the
probability is partitioned by the peak to the proportion of
$\sqrt{b}/(\sqrt{a}+\sqrt{b})$ (left) and $\sqrt{a}/(\sqrt{a}+\sqrt{b})$
(right); the functional form near the peak is controlled by the cubic term in
the Taylor expansion when $a\ne b$; 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 $e^{\frac{z-\log A}{b} }/b$ and $e^{ -\frac{z-\log A}{a}}/a$. 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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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: math
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con