Summary
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]
| Parameters |
shape (real) shape (real) scale (real) | ||
|---|---|---|---|
| Support | |||
| CDF | |||
| Mean |
where is the "probabilists' Hermite polynomials" | ||
| Median | |||
| Variance | |||
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c90900b770d8d7cd1aae2ca2debb2cc8bb8e594)
![{\displaystyle 1-e^{-\alpha \log {\frac {x}{\sigma }}-\beta [\log {\frac {x}{\sigma }}]^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d974e71cf9bb73027b65af06a6957ca23463b2ae)
Distribution edit
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[3] considered only the two-parameter model (with α = 0), with CDF
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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26a221e4b829acfbec94c6f0af041e4f7f7a728b)
Simulation edit
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 edit
- If , then X has a Pareto distribution with
- If , then , where
Software edit
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]
See also edit
References edit
- ^ Kleiber, Christian; Kotz, Samuel (2003). "Chapter 7.1: Benini Distribution". Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. ISBN 978-0-471-15064-0.
- ^ A. Sen and J. Silber (2001). Handbook of Income Inequality Measurement, Boston:Kluwer, Section 3: Personal Income Distribution Models.
- ^ a b 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.
- ^ See the references in Kleiber and Kotz (2003), p. 236.
- ^ Thomas W. Yee (2010). "The VGAM Package for Categorical Data Analysis". Journal of Statistical Software. 32 (10): 1–34. Also see the VGAM reference manual. Archived 2013-09-23 at the Wayback Machine.
External links edit
- Benini Distribution at Wolfram Mathematica (definition and plots of pdf)
