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# Dagum distribution

Parameters Probability density function Cumulative distribution function ${\displaystyle p>0}$ shape ${\displaystyle a>0}$ shape ${\displaystyle b>0}$ scale ${\displaystyle x>0}$ ${\displaystyle {\frac {ap}{x}}\left({\frac {({\tfrac {x}{b}})^{ap}}{\left(({\tfrac {x}{b}})^{a}+1\right)^{p+1}}}\right)}$ ${\displaystyle {\left(1+{\left({\frac {x}{b}}\right)}^{-a}\right)}^{-p}}$ ${\displaystyle {\begin{cases}-{\frac {b}{a}}{\frac {\Gamma \left(-{\tfrac {1}{a}}\right)\Gamma \left({\tfrac {1}{a}}+p\right)}{\Gamma (p)}}&{\text{if}}\ a>1\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}}$ ${\displaystyle b{\left(-1+2^{\tfrac {1}{p}}\right)}^{-{\tfrac {1}{a}}}}$ ${\displaystyle b{\left({\frac {ap-1}{a+1}}\right)}^{\tfrac {1}{a}}}$ ${\displaystyle {\begin{cases}-{\frac {b^{2}}{a^{2}}}\left(2a{\frac {\Gamma \left(-{\tfrac {2}{a}}\right)\,\Gamma \left({\tfrac {2}{a}}+p\right)}{\Gamma \left(p\right)}}+\left({\frac {\Gamma \left(-{\tfrac {1}{a}}\right)\Gamma \left({\tfrac {1}{a}}+p\right)}{\Gamma \left(p\right)}}\right)^{2}\right)&{\text{if}}\ a>2\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}}$

The Dagum distribution is a continuous probability distribution defined over positive real numbers. It is named after Camilo Dagum, who proposed it in a series of papers in the 1970s.[1][2] The Dagum distribution arose from several variants of a new model on the size distribution of personal income and is mostly associated with the study of income distribution. There is both a three-parameter specification (Type I) and a four-parameter specification (Type II) of the Dagum distribution; a summary of the genesis of this distribution can be found in "A Guide to the Dagum Distributions".[3] A general source on statistical size distributions often cited in work using the Dagum distribution is Statistical Size Distributions in Economics and Actuarial Sciences.[4]

## Definition

The cumulative distribution function of the Dagum distribution (Type I) is given by

${\displaystyle F(x;a,b,p)=\left(1+\left({\frac {x}{b}}\right)^{-a}\right)^{-p}{\text{ for }}x>0{\text{ where }}a,b,p>0.}$

The corresponding probability density function is given by

${\displaystyle f(x;a,b,p)={\frac {ap}{x}}\left({\frac {({\tfrac {x}{b}})^{ap}}{\left(({\tfrac {x}{b}})^{a}+1\right)^{p+1}}}\right).}$

The quantile function is given by

${\displaystyle Q(u;a,b,p)=b(u^{-1/p}-1)^{-1/a}}$

The Dagum distribution can be derived as a special case of the generalized Beta II (GB2) distribution (a generalization of the Beta prime distribution):

${\displaystyle X\sim D(a,b,p)\iff X\sim GB2(a,b,p,1)}$

There is also an intimate relationship between the Dagum and Singh–Maddala distribution.

${\displaystyle X\sim D(a,b,p)\iff {\frac {1}{X}}\sim SM(a,{\tfrac {1}{b}},p)}$

The cumulative distribution function of the Dagum (Type II) distribution adds a point mass at the origin and then follows a Dagum (Type I) distribution over the rest of the support (i.e. over the positive halfline)

${\displaystyle F(x;a,b,p,\delta )=\delta +(1-\delta )\left(1+\left({\frac {x}{b}}\right)^{-a}\right)^{-p}.}$

## Use in economics

The Dagum distribution is often used to model income and wealth distribution. The relation between the Dagum Type I and the gini coefficient is summarized in the formula below:[5]

${\displaystyle G={\frac {\Gamma (p)\Gamma (2p+1/a)}{\Gamma (2p)\Gamma (p+1/a)}}-1,}$

where ${\displaystyle \Gamma (\cdot )}$ is the gamma function. Note that this value is independent from the scale-parameter, ${\displaystyle b}$.

Although the Dagum distribution is not the only three parameter distribution used to model income distribution it is usually the most appropriate.[6]

## References

1. ^ Dagum, Camilo (1975); A model of income distribution and the conditions of existence of moments of finite order; Bulletin of the International Statistical Institute, 46 (Proceedings of the 40th Session of the ISI, Contributed Paper), 199–205.
2. ^ Dagum, Camilo (1977); A new model of personal income distribution: Specification and estimation; Economie Appliquée, 30, 413–437.
3. ^ Kleiber, Christian (2008) "A Guide to the Dagum Distributions" in Chotikapanich, Duangkamon (ed.) Modeling Income Distributions and Lorenz Curves (Economic Studies in Inequality, Social Exclusion and Well-Being), Chapter 6, Springer
4. ^ Kleiber, Christian and Samuel Kotz (2003) Statistical Size Distributions in Economics and Actuarial Sciences, Wiley
5. ^ Kleiber, Christian (2007); A Guide to the Dagum Distributions (Working paper)
6. ^ Bandourian, Ripsy (2002); A Comparison of Parametric Models for Income Distribution Across Contries and Over Time
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