Benini distribution
Parameters | shape (real) shape (real) scale (real) | ||
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Support | |||
CDF | |||
Mean | where is the "probabilists' Hermite polynomials" | ||
Median | |||
Variance |
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 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
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
Related distributions
- If , then X has a Pareto distribution with
- If , then , where
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
- Conditional probability distribution
- Joint probability distribution
- Quasiprobability distribution
- Empirical probability distribution
- Histogram
- Riemann–Stieltjes integral application to probability theory
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)
- v
- t
- e
univariate
with finite support | |
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with infinite support |
univariate
univariate
continuous- discrete |
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(joint)
- Discrete:
- Ewens
- Multinomial
- Continuous:
- Dirichlet
- Multivariate Laplace
- Multivariate normal
- Multivariate stable
- Multivariate t
- Normal-gamma
- Matrix-valued:
- LKJ
- Matrix normal
- Matrix t
- Matrix gamma
- Wishart
- 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
and singular
- Degenerate
- Dirac delta function
- Singular
- Cantor
Category
Commons