Benini distribution

Benini

Parameters	${\displaystyle \alpha >0}$ shape (real) ${\displaystyle \beta >0}$ shape (real) ${\displaystyle \sigma >0}$ scale (real)
Support	${\displaystyle x>\sigma }$
PDF	${\displaystyle e^{-\alpha \log {\frac {x}{\sigma }}-\beta \left[\log {\frac {x}{\sigma }}\right]^{2}}\left({\frac {\alpha }{x}}+{\frac {2\beta \log {\frac {x}{\sigma }}}{x}}\right)}$
CDF	${\displaystyle 1-e^{-\alpha \log {\frac {x}{\sigma }}-\beta [\log {\frac {x}{\sigma }}]^{2}}}$
Mean	${\displaystyle \sigma +{\tfrac {\sigma }{\sqrt {2\beta }}}H_{-1}\left({\tfrac {-1+\alpha }{\sqrt {2\beta }}}\right)}$ where ${\displaystyle H_{n}(x)}$ is the "probabilists' Hermite polynomials"
Median	${\displaystyle \sigma \left(e^{\frac {-\alpha +{\sqrt {\alpha ^{2}+\beta \log {16}}}}{2\beta }}\right)}$
Variance	${\displaystyle \left(\sigma ^{2}+{\tfrac {2\sigma ^{2}}{\sqrt {2\beta }}}H_{-1}\left({\tfrac {-2+\alpha }{\sqrt {2\beta }}}\right)\right)-\mu ^{2}}$

In probability, statistics, economics, and actuarial science, the Benini distribution is a continuous probability distribution that is a statistical size distribution often applied to model incomes, severity of claims or losses in actuarial applications, and other economic data.^[1][2] Its tail behavior decays faster than a power law, but not as fast as an exponential. This distribution was introduced by Rodolfo Benini in 1905.^[3] Somewhat later than Benini's original work, the distribution has been independently discovered or discussed by a number of authors.^[4]

The Benini distribution ${\displaystyle \operatorname {Benini} (\alpha ,\beta ,\sigma )}$ is a three-parameter distribution, which has cumulative distribution function (CDF)

${\displaystyle F(x)=1-\exp {\big \{}-\alpha (\log x-\log \sigma )-\beta (\log x-\log \sigma )^{2}{\big \}}=1-\left({\frac {x}{\sigma }}\right)^{-\alpha -\beta \log {\left({\frac {x}{\sigma }}\right)}},}$

where ${\displaystyle x\geq \sigma }$, shape parameters α, β > 0, and σ > 0 is a scale parameter.

For parsimony, Benini^[3] considered only the two-parameter model (with α = 0), with CDF

${\displaystyle F(x)=1-\exp {\big \{}-\beta (\log x-\log \sigma )^{2}{\big \}}=1-\left({\frac {x}{\sigma }}\right)^{-\beta (\log x-\log \sigma )}.}$

The density of the two-parameter Benini model is

${\displaystyle f(x)={\frac {2\beta }{x}}\exp \left\{-\beta \left[\log \left({\frac {x}{\sigma }}\right)\right]^{2}\right\}\log \left({\frac {x}{\sigma }}\right),\quad x\geq \sigma >0.}$

A two-parameter Benini variable can be generated by the inverse probability transform method. For the two-parameter model, the quantile function (inverse CDF) is

${\displaystyle F^{-1}(u)=\sigma \exp {\sqrt {-{\frac {1}{\beta }}\log(1-u)}},\quad 0<u<1.}$

• If ${\displaystyle X\sim \operatorname {Benini} (\alpha ,0,\sigma )}$, then X has a Pareto distribution with ${\displaystyle x_{\text{m}}=\sigma .}$
• If ${\displaystyle X\sim \operatorname {Benini} (0,{\tfrac {1}{2\sigma ^{2}}},1)}$, then ${\displaystyle X\sim e^{U}}$, where ${\displaystyle U\sim \operatorname {Rayleigh} (\sigma ).}$

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The two-parameter Benini distribution density, probability distribution, quantile function and random-number generator are implemented in the VGAM package for R, which also provides maximum-likelihood estimation of the shape parameter.^[5]

• Conditional probability distribution
• Joint probability distribution
• Quasiprobability distribution
• Empirical probability distribution
• Histogram
• Riemann–Stieltjes integral application to probability theory

• Benini Distribution at Wolfram Mathematica (definition and plots of pdf)