Dagum distribution

Dagum Distribution
Probability density function
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	$\begin{cases} -\frac{b^2}{a^2} \left(2 a \frac{\Gamma\left(-\tfrac{2}{a}\right) \, \Gamma\left(\tfrac{2}{a} + p\right)}{\Gamma\left(p\right)} + \left( \frac{\Gamma\left(-\tfrac{1}{a}\right) \Gamma\left(\tfrac{1}{a} + p\right)}{\Gamma\left(p\right)} \right)^2\right) & \text{if}\ a>2 \\ \text{Indeterminate} & \text{otherwise}\ \end{cases}$

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

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 quantile function is given by

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):

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 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} .

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]

G=\frac{\Gamma (p) \Gamma (2p+1/a)}{\Gamma (2p) \Gamma (p+1/a)}-1

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

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

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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