# Dagum distribution

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]

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

## 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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