# Dagum distribution

Parameters Probability density function Cumulative distribution function $p>0$ shape $a>0$ shape $b>0$ scale $x>0$ ${\frac {ap}{x}}\left({\frac {({\tfrac {x}{b}})^{ap}}{\left(({\tfrac {x}{b}})^{a}+1\right)^{p+1}}}\right)$ ${\left(1+{\left({\frac {x}{b}}\right)}^{-a}\right)}^{-p}$ ${\begin{cases}-{\frac {b}{a}}{\frac {\Gamma \left(-{\tfrac {1}{a}}\right)\Gamma \left({\tfrac {1}{a}}+p\right)}{\Gamma (p)}}&{\text{if}}\ a>1\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}$ $b{\left(-1+2^{\tfrac {1}{p}}\right)}^{-{\tfrac {1}{a}}}$ $b{\left({\frac {ap-1}{a+1}}\right)}^{\tfrac {1}{a}}$ ${\begin{cases}-{\frac {b^{2}}{a^{2}}}\left(2a{\frac {\Gamma \left(-{\tfrac {2}{a}}\right)\,\Gamma \left({\tfrac {2}{a}}+p\right)}{\Gamma \left(p\right)}}+\left({\frac {\Gamma \left(-{\tfrac {1}{a}}\right)\Gamma \left({\tfrac {1}{a}}+p\right)}{\Gamma \left(p\right)}}\right)^{2}\right)&{\text{if}}\ a>2\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}$ 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. 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". A general source on statistical size distributions often cited in work using the Dagum distribution is Statistical Size Distributions in Economics and Actuarial Sciences.

## Definition

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

$F(x;a,b,p)=\left(1+\left({\frac {x}{b}}\right)^{-a}\right)^{-p}{\text{ for }}x>0{\text{ where }}a,b,p>0.$ The corresponding probability density function is given by

$f(x;a,b,p)={\frac {ap}{x}}\left({\frac {({\tfrac {x}{b}})^{ap}}{\left(({\tfrac {x}{b}})^{a}+1\right)^{p+1}}}\right).$ The quantile function is given by

$Q(u;a,b,p)=b(u^{-1/p}-1)^{-1/a}$ The Dagum distribution can be derived as a special case of the generalized Beta II (GB2) distribution (a generalization of the Beta prime distribution):

$X\sim D(a,b,p)\iff X\sim GB2(a,b,p,1)$ There is also an intimate relationship between the Dagum and Singh–Maddala distribution.

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

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

$G={\frac {\Gamma (p)\Gamma (2p+1/a)}{\Gamma (2p)\Gamma (p+1/a)}}-1,$ where $\Gamma (\cdot )$ is the gamma function. Note that this value is independent from the scale-parameter, $b$ .

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