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
\operatorname{Benini}(\alpha,\beta,\sigma)
F(x)=1-\exp\{-\alpha(logx-log\sigma)-\beta(logx-log\sigma)2\} =1-\left(
| x | |
| \sigma |
| ||||||
| \right) |
x\geq\sigma
For parsimony, Benini[3] considered only the two-parameter model (with α = 0), with CDF
F(x)=1-\exp\{-\beta(logx-log\sigma)2\}=1-\left(
| x | |
| \sigma |
\right)-\beta(log.
f(x)=
| 2\beta | \exp\left\{-\beta\left[log\left( | |
| x |
| x | |
| \sigma |
\right)\right]2\right\} log\left(
| x | |
| \sigma |
\right), 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
F-1(u)=\sigma\exp\sqrt{-
| 1 | |
| \beta |
log(1-u)}, 0<u<1.
X\sim\operatorname{Benini}(\alpha,0,\sigma)
xm=\sigma.
X\sim\operatorname{Benini}(0,\tfrac{1}{2\sigma2},1)
X\simeU
U\sim\operatorname{Rayleigh}(\sigma).
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]