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Benini distribution

Continuous probability distribution

Summary

Continuous probability distribution

name =Benini| type =density| pdf_image = | cdf_image = | parameters =\alpha0 shape (real) \beta0 shape (real) \sigma0 scale (real)| support =x\sigma | pdf =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^{-\alpha\log{\frac{x}{\sigma}}-\beta[\log{\frac{x}{\sigma}}]^2}| mean =\sigma+\tfrac{\sigma}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-1+\alpha}{\sqrt{2\beta}}\right)
where H_n(x) is the "probabilists' Hermite polynomials"| median =\sigma \left(e^{\frac{-\alpha+\sqrt{\alpha^2+\beta\log{16}}}{2\beta}}\right)| mode =| variance = \left(\sigma^2+\tfrac{2\sigma^2}{\sqrt{2\beta}} H_{-1}\left(\tfrac{-2+\alpha}{\sqrt{2\beta}}\right)\right)-\mu^2 | skewness =| kurtosis =| entropy =| mgf =| char =|

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.{{cite book|title= Statistical Size Distributions in Economics and Actuarial Sciences |first1=Christian |last1=Kleiber| first2= Samuel |last2= Kotz

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

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.

References

References

  1. A. Sen and J. Silber (2001). ''Handbook of Income Inequality Measurement'', Boston:Kluwer, Section 3: Personal Income Distribution Models.
  2. 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.
  3. See the references in Kleiber and Kotz (2003), p. 236.
  4. Thomas W. Yee. (2013-09-23). ["The VGAM Package for Categorical Data Analysis"](http://www.jstatsoft.org/v32/i10/}} Also see the [http://probability.ca/cran/web/packages/VGAM/VGAM.pdf VGAM reference manual]. {{Webarchive). Journal of Statistical Software.
Wikipedia Source

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