Benini distribution

Benini
Parameters	$\alpha >0$ shape (real) $\beta >0$ shape (real) $\sigma >0$ 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	$\sigma \left(e^{\frac {-\alpha +{\sqrt {\alpha ^{2}+\beta \log {16}}}}{2\beta }}\right)$
Variance	$\left(\sigma ^{2}+{\tfrac {2\sigma ^{2}}{\sqrt {2\beta }}}H_{-1}\left({\tfrac {-2+\alpha }{\sqrt {2\beta }}}\right)\right)-\mu ^{2}$

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.^[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 $\mathrm {Benini} (\alpha ,\beta ,\sigma )$ is a three parameter distribution, which has cumulative distribution function (cdf)

F(x)=1-\exp\{-\alpha (\log x-\log \sigma )-\beta (\log x-\log \sigma )^{2}\}=1-\left({\frac {x}{\sigma }}\right)^{-\alpha -\beta \log {\left({\frac {x}{\sigma }}\right)}}

where $x\geq \sigma$ , shape parametes 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\{-\beta (\log x-\log \sigma )^{2}\}=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\}\cdot \log \left({\frac {x}{\sigma }}\right),\qquad 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 $X\sim \mathrm {Benini} (\alpha ,0,\sigma )\,$ , then X has a Pareto distribution with $x_{\mathrm {m} }=\sigma$
If $X\sim \mathrm {Benini} (0,{\tfrac {1}{2\sigma ^{2}}},1),$ then $X\sim e^{U}$ where $U\sim \mathrm {Rayleigh} (\sigma )$

Software

The (two parameter) Benini distribution density, probability distribution, quantile function and random number generator is 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.
1 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.
↑ 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.

External links

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

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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