Benini distribution

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

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

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

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

The density of the two-parameter Benini model is

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

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

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

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