# Dagum Distribution

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The Dagum distribution was proposed by Camilo Dagum in the 1970s as an alternative to two popular models, the Pareto Distribution and log-normal distribution. The goal was to develop a model that could better handle heavy-tailed data when applied to income and wealth distributions. Let’s take a closer look at what the Dagum Distribution is and why it matters.

## The Basics of the Dagum Distribution

The Dagum Distribution (also called the Mielke Beta-Kappa distribution or inverse Burr distribution) is a three-parameter model that has been used to explain income inequality in countries around the world. It is named after its creator, Argentine statistician and econometrician Camilo Dagum [1], who developed it with the intention of improving upon existing models for predicting income distributions.

Specifically, Dagum wanted to address issues with how existing models handled heavy-tailed data often found in empirical income and wealth distributions while still allowing for an interior mode (a mode at zero). Issues include:

• The Pareto distribution handles heavy tails well, but the log-normal does not.
• The log-normal distribution allows an interior mode, the Pareto does not.
• Dagum’s distribution allows for both an interior mode and it heavy tails [2].

The Dagum distribution often performs better than other two/three parameter income/wealth distribution models when applied to empirical data [3]. It is therefore somewhat surprising that the model isn’t as popular as other models such as the Pareto.

## Properties

A three-parameter (Type I) Dagum distribution has parameters λ, β, and δ. which determine where the peak of the curve will be located (the mode) and how quickly it declines on either side (heavy tails). In addition, these parameters also determine how skewed or flat the curve will be (asymmetry). As such, they can be used to accurately predict certain economic phenomena such as economic inequality across different countries or regions.

The Dagum distribution probability density function (Type I) is

Where:

When β = 1, the distribution is the log-logistic distribution.

The three parameter Dagum Type I distribution evolved from Dagum’s experimentation with a shifted log-logistic distribution [4]. Two four-parameter (Type II) generalizations were also developed.

The Dagum distribution can be derived as a special case of the generalized Beta II (GB2) distribution; it is also the inverse of the Burr distribution.

## Real World Applications of the Dagum Distribution

Since its development in 1974, researchers have used the Dagum distribution to analyze various types of data related to income inequality and poverty levels across different regions. For example, many researchers have studied how changes in GDP per capita affect poverty levels using this distribution model. Other researchers have used it to study regional disparities in wages within countries or even global trends in wealth distribution between nations. Additionally, many economists use this model when making predictions about future economic growth or planning policy related to taxation or welfare spending.

## References

Image: Rafael Marcondes, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons

[1] Dagum, C., A New Model of Personal Income Distribution: Specification and Estimation, Economie Applique’e, 30, 413 – 437, (1977)

[2] Kleiber, C. (2007). A Guide to the Dagum Distributions. Retrieved April 5, 2023 from: http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=BEE8C9083715B6042596B0DC170BFDA1?doi=10.1.1.610.8346&rep=rep1&type=pdf

[3] Bandourian, Ripsy; et al. (2002). “A Comparison of Parametric Models of Income Distribution Across Countries and Over Time”. Luxembourg Income Study Working Paper No. 305. SSRN 324900.

[4] Chotikapanich, D. (Ed.) (2008). Modeling Income Distributions and Lorenz Curves. Springer Science & Business Media.

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