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Dagum distribution

What it describes

The Dagum distribution is a flexible positive, right-skewed model. Its three parameters separate scale, near-origin concentration, and tail behaviour, which is useful for size and income data.

Its support is positive. For some shape values, the mean, variance, or higher moments do not exist; that possibility is part of the model rather than a numerical failure.

History and names

Camilo Dagum proposed the family in the 1970s for personal-income distributions. In economic literature it is also identified as Burr type III.

How it connects to other distributions

It belongs to the generalized beta type II family and is closely related to Burr, Beta prime, loglogistic, and Singh–Maddala models. A Beta-prime transformation helps explain its tail shape.

Where it appears

  • income, wealth, and household-size distributions
  • reliability, particle size, and strongly skewed positive measurements

Fitting cautions

The tail can dominate the mean; when comparing groups, report medians and quantiles as well as averages to avoid an extreme-sensitive reading.

A curve built to see the whole income scale

In 1977 Camilo Dagum proposed a distribution able to describe both concentration among low incomes and a heavy upper tail. In economics it is also called Burr type III, although the exact convention must be checked. Its reciprocal is closely related to Burr XII or Singh Maddala.

The family offers manageable expressions for quantiles and Lorenz curves, making it useful for studying inequality as well as density. A good income fit does not stabilize every social measure: the lower part may be affected by zeros, underreporting, or household definitions, while a few values dominate the tail. Comparing quantile shares and tail index gives a more substantive assessment than overlaying histograms.

Decision guide

A good candidate when: income, size, or duration data are positive, with substantial lower-end mass and a heavy upper tail.

Compare it with: Burr XII, Loglogistic, and Beta Prime. Evaluate Lorenz curves or quantiles as well as density; they expose economically important differences better than a central histogram.

References

  • SciPy reference: scipy.stats.mielke — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.
  • Dagum, C. (1977). A new model of personal income distribution: Specification and estimation. Swiss Journal of Economics and Statistics, 113, 185–211.
  • Kleiber, C. & Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley.

Dagum Distribution: equations and calculator

Distribution defintion

X∼Dagum(a,b,p)X\sim\mathrm{Dagum}\left(a,b,p\right)

Distribution domain

x∈(0,∞)x\in\left(0,\infty\right)

Parameters domain and parameters constraints

a∈R+,b∈R+,p∈R+a\in\mathbb{R}^{+},b\in\mathbb{R}^{+},p\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=(1+(xb)−a)−pF_{X}\left(x\right)={\left(1+{\left(\frac{x}{b}\right)}^{-a}\right)}^{-p}

Probability density function

fX(x)=apx((xb)ap((xb)a+1)p+1)f_{X}\left(x\right)=\frac{a p}{x}\left(\frac{(\tfrac{x}{b})^{a p}}{\left((\tfrac{x}{b})^a+1\right)^{p+1}}\right)

Percent point function/Sample

FX−1(u)=b(u−1/p−1)−1/aF^{-1}_{X}\left(u\right)=b(u^{-1/p}-1)^{-1/a}

Non-central parametric moments

μk′=E[Xk]=∫0∞xkfX(x)dx=pbk⋅Beta(ap+ka,a−ka)\mu'_{k}=E[X^k]=\int_{0}^{\infty}x^{k}f_{X}\left(x\right)dx=pb^{k}\cdot \text{Beta}\left(\frac{ap+k}{a},\frac{a-k}{a}\right)

Parametric mean

Mean(X)=μ1′\mathrm{Mean}(X)=\mu'_{1}

Parametric variance

Variance(X)=μ2′−μ1′2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}

Parametric median

Median(X)=b(−1+21p)−1a\mathrm{Median}(X)=b{\left(-1+2^{\tfrac{1}{p}}\right)}^{-\tfrac{1}{a}}

Parametric mode

Mode(X)=b(ap−1a+1)1a\mathrm{Mode}(X)=b{\left(\frac{ap-1}{a+1}\right)}^{\tfrac{1}{a}}

Additional information and definitions

b:Scale parameterb:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}