Battle of Locust Grove

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]

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^[3] 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<u<1.

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.^[5]

References

^ 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

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

[1] 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.

[2] A. Sen and J. Silber (2001). Handbook of Income Inequality Measurement, Boston:Kluwer, Section 3: Personal Income Distribution Models.

[RB1905-3] 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.

[4] See the references in Kleiber and Kotz (2003), p. 236.

[5] 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.

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