Dagum distribution

The Gini coefficient for a continuous probability distribution takes the form:

${\displaystyle G={1 \over {\mu }}\int _{0}^{\infty }F(1-F)dx}$

where ${\displaystyle F}$ is the CDF of the distribution and ${\displaystyle \mu }$ is the expected value. For the Dagum distribution, assuming ${\displaystyle a>1}$, the formula for the Gini coefficient becomes:

${\displaystyle G={\frac {\Gamma (p)}{b\Gamma (1-1/a)\Gamma (p+1/a)}}\int _{0}^{\infty }\left[1+\left({\frac {x}{b}}\right)^{-a}\right]^{-p}\left\{1-\left[1+\left({\frac {x}{b}}\right)^{-a}\right]^{-p}\right\}dx}$

Defining the substitution ${\displaystyle u=(x/b)^{-a}}$ leads to the simpler equation:

${\displaystyle G={\frac {\Gamma (p)}{a\Gamma (1-1/a)\Gamma (p+1/a)}}\int _{0}^{\infty }u^{-1/a-1}\left(1+u\right)^{-p}\left[1-\left(1+u\right)^{-p}\right]du}$

And making the substitution ${\displaystyle z=u/(1+u)}$ further changes the Gini coefficient formula to:

${\displaystyle G={\frac {\Gamma (p)}{a\Gamma (1-1/a)\Gamma (p+1/a)}}\int _{0}^{1}z^{-1/a-1}\left(1-z\right)^{p+1/a-1}\left[1-\left(1-z\right)^{p}\right]dz}$

Each of the integral components is equivalent to the standard beta function, which may be written in terms of the gamma function:

${\displaystyle {\text{B}}(x,y)={\Gamma (x)\Gamma (y) \over {\Gamma (x+y)}}}$

Then using this definition, and the fact that ${\displaystyle a\Gamma (1-1/a)=-\Gamma (-1/a)}$, the Gini coefficient may be written as:

${\displaystyle {\begin{aligned}G&={\frac {\Gamma (p)}{\Gamma (-1/a)\Gamma (p+1/a)}}\left[{\frac {\Gamma (-1/a)\Gamma (2p+1/a)}{\Gamma (2p)}}-{\frac {\Gamma (-1/a)\Gamma (p+1/a)}{\Gamma (p)}}\right]\\&={\frac {\Gamma (p)\Gamma (2p+1/a)}{\Gamma (2p)\Gamma (p+1/a)}}-1\end{aligned}}}$

with the last term being the Gini coefficient for the Dagum distribution.