Dagum distribution

Dagum Distribution
	Probability density function
	Cumulative distribution function
Parameters	shape ; shape ; scale
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance

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.^[2]^[3] 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".^[4] A general source on statistical size distributions often cited in work using the Dagum distribution is Statistical Size Distributions in Economics and Actuarial Sciences.^[5]

^ 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.

[ChotikapanichetalEconometrics2018-1] Chotikapanich, Duangkamon; et al. (2018). "Using the GB2 Income Distribution". Econometrics. 6 (2): 21. doi:10.3390/econometrics6020021. hdl:10419/195459.

[dagum1975-2] 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.

[dagum1977-3] Dagum, Camilo (1977). "A new model of personal income distribution: Specification and estimation". Économie Appliquée. 30: 413–437.

[kleiber2008-4] 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.

[kleiber2003-5] Kleiber, Christian; Kotz, Samuel (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Wiley. ISBN 0-471-15064-9.

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