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Benini distribution

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. 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. Somewhat later than Benini's original work, the distribution has been independently discovered or discussed by a number of authors.

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Benini
Parameters α > 0 {\displaystyle \alpha >0} shape (real)
β > 0 {\displaystyle \beta >0} shape (real)
σ > 0 {\displaystyle \sigma >0} scale (real)
Support x > σ {\displaystyle x>\sigma }
PDF e − α log ⁡ x σ − β [ log ⁡ x σ ] 2 ( α x + 2 β log ⁡ x σ x ) {\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)}
CDF 1 − e − α log ⁡ x σ − β [ log ⁡ x σ ] 2 {\displaystyle 1-e^{-\alpha \log {\frac {x}{\sigma }}-\beta [\log {\frac {x}{\sigma }}]^{2}}}
Mean σ + σ 2 β H − 1 ( − 1 + α 2 β ) {\displaystyle \sigma +{\tfrac {\sigma }{\sqrt {2\beta }}}H_{-1}\left({\tfrac {-1+\alpha }{\sqrt {2\beta }}}\right)}
where H n ( x ) {\displaystyle H_{n}(x)} is the "probabilists' Hermite polynomials"
Median σ ( e − α + α 2 + β log ⁡ 16 2 β ) {\displaystyle \sigma \left(e^{\frac {-\alpha +{\sqrt {\alpha ^{2}+\beta \log {16}}}}{2\beta }}\right)}
Mode σ e 8 β + 1 − 2 α − 1 4 β {\displaystyle \sigma e^{\frac {{\sqrt {8\beta +1}}-2\alpha -1}{4\beta }}} if β > α ( α + 1 ) / 2 {\displaystyle \beta >\alpha (\alpha +1)/2} , otherwise σ {\displaystyle \sigma }
Variance ( σ 2 + 2 σ 2 2 β H − 1 ( − 2 + α 2 β ) ) − μ 2 {\displaystyle \left(\sigma ^{2}+{\tfrac {2\sigma ^{2}}{\sqrt {2\beta }}}H_{-1}\left({\tfrac {-2+\alpha }{\sqrt {2\beta }}}\right)\right)-\mu ^{2}}

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.12 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 Benini ⁡ ( α , β , σ ) {\displaystyle \operatorname {Benini} (\alpha ,\beta ,\sigma )} is a three-parameter distribution, which has cumulative distribution function (CDF)

F ( x ) = 1 − exp ⁡ { − α ( log ⁡ x − log ⁡ σ ) − β ( log ⁡ x − log ⁡ σ ) 2 } = 1 − ( x σ ) − α − β log ⁡ ( x σ ) , {\displaystyle 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 ≥ σ {\displaystyle x\geq \sigma } , shape parameters α, β > 0, and σ > 0 is a scale parameter.

For parsimony, Benini3 considered only the two-parameter model (with α = 0), with CDF

F ( x ) = 1 − exp ⁡ { − β ( log ⁡ x − log ⁡ σ ) 2 } = 1 − ( x σ ) − β ( log ⁡ x − log ⁡ σ ) . {\displaystyle 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 ) = 2 β x exp ⁡ { − β [ log ⁡ ( x σ ) ] 2 } log ⁡ ( x σ ) , x ≥ σ > 0. {\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.}

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 ) = σ exp ⁡ − 1 β log ⁡ ( 1 − u ) , 0 < u < 1. {\displaystyle F^{-1}(u)=\sigma \exp {\sqrt {-{\frac {1}{\beta }}\log(1-u)}},\quad 0<u<1.}
  • If X ∼ Benini ⁡ ( α , 0 , σ ) {\displaystyle X\sim \operatorname {Benini} (\alpha ,0,\sigma )} , then X has a Pareto distribution with x m = σ . {\displaystyle x_{\text{m}}=\sigma .}
  • If X ∼ Benini ⁡ ( 0 , 1 2 σ 2 , 1 ) {\displaystyle X\sim \operatorname {Benini} (0,{\tfrac {1}{2\sigma ^{2}}},1)} , then X ∼ e U {\displaystyle X\sim e^{U}} , where U ∼ Rayleigh ⁡ ( σ ) . {\displaystyle 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

See also

See also

References

References

  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.
  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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