Dagum distribution

Dagum Distribution

Probability density function
Cumulative distribution function No image available
parameters:	p > 0 shape a > 0 shape b > 0 scale
support:	x > 0
pdf:	$\frac{a p}{x} \left( \frac{(\tfrac{x}{b})^{a p}}{\left((\tfrac{x}{b})^a + 1 \right)^{p+1}} \right)$
cdf:	${\left( 1+{\left(\frac{x}{b}\right)}^{-a} \right)}^{-p}$
mean:	$\begin{cases} -\frac{b}{a}\frac{\Gamma\left(-\tfrac{1}{a}\right)\Gamma\left(\tfrac{1}{a}+p\right)}{\Gamma(p)} & \text{if}\ a>1 \\ \text{Indeterminate} & \text{otherwise}\ \end{cases}$
median:	$b{\left(-1+2^{\tfrac{1}{p}}\right)}^{-\tfrac{1}{a}}$
mode:	$b{\left( \frac{ap-1}{a+1} \right)}^{\tfrac{1}{a}}$
variance:	$\begin{cases} -\frac{b^2}{a^2} \left(2 a \frac{\Gamma\left(-\tfrac{2}{a}\right) \, \Gamma\left(\tfrac{2}{a} + p\right)}{\Gamma\left(p\right)} + \left( \frac{\Gamma\left(-\tfrac{1}{a}\right) \Gamma\left(\tfrac{1}{a} + p\right)}{\Gamma\left(p\right)} \right)^2\right) & \text{if}\ a>2 \\ \text{Indeterminate} & \text{otherwise}\ \end{cases}$
mgf:	UNIQ21c15a213ff61bcc-math-0000000A-QINU
cf:	UNIQ21c15a213ff61bcc-math-0000000B-QINU

The Dagum distribution is a continuous probability distribution defined over all 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.^[3] A general source on statistical size distributions often cited in work using the Dagum distribution is.^[4]

Definition

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

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

The corresponding probability density function is given by

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

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

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

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 (Proceeding 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

External links

• Camilo Dagum (1925 - 2005) : obituary