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.A. Sen and J. Silber (2001). Handbook of Income Inequality Measurement, Boston:Kluwer, Section 3: Personal Income Distribution Models. 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.Benini, R. (1905). I diagrammi a scala logaritmica (a proposito della graduazione per valore delle successioni ereditarie in Italia, Francia e Inghilterra). Giornale degli Economisti, Series II, 16, 222–231. Somewhat later than Benini's original work, the distribution has been independently discovered or discussed by a number of authors.See the references in Kleiber and Kotz (2003), p. 236.
Distribution
The Benini distribution is a three-parameter distribution, which has cumulative distribution function (CDF)
where , 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
The density of the two-parameter Benini model is
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
Related distributions
If , then X has a Pareto distribution with If , then , where
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. Also see the VGAM reference manual. .
See also Conditional probability distribution Joint probability distribution Quasiprobability distribution Empirical probability distribution Histogram Riemann–Stieltjes integral application to probability theory
References
External links Benini Distribution at Wolfram Mathematica (definition and plots of pdf)