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Dagum Distribution
Probability density function
Cumulative distribution function
Parameters	$p>0$ shape $a>0$ shape $b>0$ scale
Support	$x>0$
PDF	${\frac {ap}{x}}\left({\frac {({\tfrac {x}{b}})^{ap}}{\left(({\tfrac {x}{b}})^{a}+1\right)^{p+1}}}\right)$
CDF	${\left(1+{\left({\frac {x}{b}}\right)}^{-a}\right)}^{-p}$
Quantile	$b(u^{-1/p}-1)^{-1/a}$
Mean	${\begin{cases}b{\frac {\Gamma \left(1-{\tfrac {1}{a}}\right)\Gamma \left(p+{\tfrac {1}{a}}\right)}{\Gamma (p)}}&{\text{if}}\ a>1\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}$
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	${\begin{cases}{b^{2}}\left({\frac {\Gamma \left(1-{\tfrac {2}{a}}\right)\,\Gamma \left(p+{\tfrac {2}{a}}\right)}{\Gamma \left(p\right)}}-\left({\frac {\Gamma \left(1-{\tfrac {1}{a}}\right)\Gamma \left(p+{\tfrac {1}{a}}\right)}{\Gamma \left(p\right)}}\right)^{2}\right)&{\text{if}}\ a>2\\{\text{Indeterminate}}&{\text{otherwise}}\ \end{cases}}$

The Dagum distribution (or Mielke Beta-Kappa 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. 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". A general source on statistical size distributions often cited in work using the Dagum distribution is Statistical Size Distributions in Economics and Actuarial Sciences.

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}{\text{ for }}x>0{\text{ where }}a,b,p>0.

The corresponding probability density function is given by

f(x;a,b,p)={\frac {ap}{x}}\left({\frac {({\tfrac {x}{b}})^{ap}}{\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}={\frac {bu^{\frac {1}{pa}}}{\left(1-u^{\frac {1}{p}}\right)^{\frac {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 / Burr 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}.

The quantile function of the Dagum (Type II) distribution is

Q(u;a,b,p,\delta )=b\left^{-{\frac {1}{a}}}.

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Use in economics

The Dagum distribution is 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:

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

where $\Gamma (\cdot )$ is the gamma function. Note that this value is independent from the scale-parameter, $b$ .

Derivation of the Gini coefficient for the Dagum distribution
The Gini coefficient for a continuous probability distribution takes the form: $G={1 \over {\mu }}\int _{0}^{\infty }F(1-F)dx$ where $F$ is the CDF of the distribution and $\mu$ is the expected value. For the Dagum distribution, assuming $a>1$ , the formula for the Gini coefficient becomes: $G={\frac {\Gamma (p)}{b\Gamma (1-1/a)\Gamma (p+1/a)}}\int _{0}^{\infty }\left^{-p}\left\{1-\left^{-p}\right\}dx$ Defining the substitution $u=(x/b)^{-a}$ leads to the simpler equation: $G={\frac {\Gamma (p)}{a\Gamma (1-1/a)\Gamma (p+1/a)}}\int _{0}^{\infty }u^{-1/a-1}\left(1+u\right)^{-p}\leftdu$ And making the substitution $z=u/(1+u)$ further changes the Gini coefficient formula to: $G={\frac {\Gamma (p)}{a\Gamma (1-1/a)\Gamma (p+1/a)}}\int _{0}^{1}z^{-1/a-1}\left(1-z\right)^{p+1/a-1}\leftdz$ Each of the integral components is equivalent to the standard beta function, which may be written in terms of the gamma function: ${\text{B}}(x,y)={\Gamma (x)\Gamma (y) \over {\Gamma (x+y)}}$ Then using this definition, and the fact that $a\Gamma (1-1/a)=-\Gamma (-1/a)$ , the Gini coefficient may be written as: ${\begin{aligned}G&={\frac {\Gamma (p)}{\Gamma (-1/a)\Gamma (p+1/a)}}\left\\&={\frac {\Gamma (p)\Gamma (2p+1/a)}{\Gamma (2p)\Gamma (p+1/a)}}-1\end{aligned}}$ with the last term being the Gini coefficient for the Dagum distribution.

Derivation of the Gini coefficient for the Dagum distribution

The Gini coefficient for a continuous probability distribution takes the form:

G={1 \over {\mu }}\int _{0}^{\infty }F(1-F)dx

where $F$ is the CDF of the distribution and $\mu$ is the expected value. For the Dagum distribution, assuming $a>1$ , the formula for the Gini coefficient becomes:

G={\frac {\Gamma (p)}{b\Gamma (1-1/a)\Gamma (p+1/a)}}\int _{0}^{\infty }\left^{-p}\left\{1-\left^{-p}\right\}dx

Defining the substitution $u=(x/b)^{-a}$ leads to the simpler equation:

G={\frac {\Gamma (p)}{a\Gamma (1-1/a)\Gamma (p+1/a)}}\int _{0}^{\infty }u^{-1/a-1}\left(1+u\right)^{-p}\leftdu

And making the substitution $z=u/(1+u)$ further changes the Gini coefficient formula to:

G={\frac {\Gamma (p)}{a\Gamma (1-1/a)\Gamma (p+1/a)}}\int _{0}^{1}z^{-1/a-1}\left(1-z\right)^{p+1/a-1}\leftdz

Each of the integral components is equivalent to the standard beta function, which may be written in terms of the gamma function:

{\text{B}}(x,y)={\Gamma (x)\Gamma (y) \over {\Gamma (x+y)}}

Then using this definition, and the fact that $a\Gamma (1-1/a)=-\Gamma (-1/a)$ , the Gini coefficient may be written as:

{\begin{aligned}G&={\frac {\Gamma (p)}{\Gamma (-1/a)\Gamma (p+1/a)}}\left\\&={\frac {\Gamma (p)\Gamma (2p+1/a)}{\Gamma (2p)\Gamma (p+1/a)}}-1\end{aligned}}

with the last term being the Gini coefficient for the Dagum distribution.

Although the Dagum distribution is not the only three-parameter distribution used to model income distribution, one study found it to usually be a better fit than other three-parameter models.

The Dagum distribution has been extended to model net wealth distribution, accounting for the observed frequencies of negative and null net wealth. This generalized model, known as the Dagum General Model of Net Wealth Distribution, is a mixture model consisting of an atomic distribution at zero (representing economic units with no wealth) with two continuous distributions for negative and positive net wealth.

References

Chotikapanich, Duangkamon; et al. (2018). "Using the GB2 Income Distribution". Econometrics. 6 (2): 21. doi:10.3390/econometrics6020021. hdl:10419/195459.
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 (Proceedings of the 40th Session of the ISI, Contributed Paper): 199–205.
Dagum, Camilo (1977). "A new model of personal income distribution: Specification and estimation". Économie Appliquée. 30: 413–437.
Kleiber, Christian (2008). "A Guide to the Dagum Distributions" (PDF). In Chotikapanich, Duangkamon (ed.). Modeling Income Distributions and Lorenz Curves. Economic Studies in Inequality, Social Exclusion and Well-Being. Springer. pp. 97–117. doi:10.1007/978-0-387-72796-7_6. ISBN 978-0-387-72756-1.
Kleiber, Christian; Kotz, Samuel (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. ISBN 0-471-15064-9.
Kleiber, Christian (2007). "A Guide to the Dagum Distributions". Working Paper.
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.
Dagum, Camilo (2006). "Wealth distribution models: analysys and applications". Statistica. 66 (3): 235–268. doi:10.6092/issn.1973-2201/1243. ISSN 1973-2201.

External links

Camilo Dagum (1925 - 2005) Archived 2017-08-15 at the Wayback Machine : obituary

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda

Mixed
univariate

continuous- discrete	Rectified Gaussian

Multivariate
(joint)

Discrete:
Ewens
Multinomial
- Dirichlet
- Negative
Continuous:
Dirichlet
- Generalized
Multivariate Laplace
Multivariate normal
Multivariate stable
Multivariate t
Normal-gamma
- Inverse
Matrix-valued:
LKJ
Matrix beta
Matrix normal
Matrix t
Matrix gamma
- Inverse
Wishart
- Normal
- Inverse
- Normal-inverse
- Complex

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

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