Benini distribution

The Benini distribution \operatorname{Benini}(\alpha, \beta, \sigma) is a three-parameter distribution, which has cumulative distribution function (CDF) : 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 x \geq \sigma, shape parameters α, β > 0, and σ > 0 is a scale parameter. For parsimony, Benini considered only the two-parameter model (with α = 0), with CDF : 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 : 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. ==Simulation==

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{-\frac{1}{\beta} \log(1 - u)}, \quad 0 ==Related distributions==

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

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. ==See also==