# Dagum distribution

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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]}

## Definition

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

- for and where

The corresponding probability density function is given by

The quantile function is given by

The Dagum distribution can be derived as a special case of the Generalized Beta II (GB2) distribution (a generalization of the Beta prime distribution):

There is also an intimate relationship between the Dagum and Singh-Maddala distribution.

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)

## Use in economics

The Dagum distribution if 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]}

Note that this value is independent on the scale-parameter, .

Although the Dagum distribution is not the only three parameter distribution used to model income distribution it is usually the most appropriate.^{[6]}

## References

- ↑ 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. - ↑ Dagum, Camilo (1977); A new model of personal income distribution: Specification and estimation;
*Economie Appliquée*, 30, 413-437. - ↑ 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 - ↑ Kleiber, Christian and Samuel Kotz (2003)
*Statistical Size Distributions in Economics and Actuarial Sciences*, Wiley - ↑ Kleiber, Christian (2007); A Guide to the Dagum Distributions (Working paper)
- ↑ Bandourian, Ripsy (2002); A Comparison of Parametric Models for Income Distribution Across Contries and Over Time

## External links

- Camilo Dagum (1925 - 2005) : obituary