Dagum Distribution

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

The Dagum distribution (or Mielke Beta-Kappa 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.^[2][3] 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".^[4] A general source on statistical size distributions often cited in work using the Dagum distribution is Statistical Size Distributions in Economics and Actuarial Sciences.^[5]

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 / Burr 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)

${\displaystyle F(x;a,b,p,\mathrm{\delta <!--\; \delta \; -->})=\mathrm{\delta <!--\; \delta \; -->}+(1-<!--\; -\; -->\mathrm{\delta <!--\; \delta \; -->}){(1+{\left(\frac{x}{b}\right)}^{-<!--\; -\; -->a}}^{}}$Zdroj:https://en.wikipedia.org?pojem=Dagum_distribution
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