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Burr Type XII
Probability density function 👁 Image
Cumulative distribution function 👁 Image
Parameters	👁 {\displaystyle c>0\!} 👁 {\displaystyle k>0\!}
Support	👁 {\displaystyle x>0\!}
PDF	👁 {\displaystyle ck{\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\!}
CDF	👁 {\displaystyle 1-\left(1+x^{c}\right)^{-k}}
Quantile	👁 {\displaystyle \lambda \left({\frac {1}{(1-U)^{\frac {1}{k}}}}-1\right)^{\frac {1}{c}}}
Mean	👁 {\displaystyle \mu _{1}=k\operatorname {\mathrm {B} } (k-1/c,\,1+1/c)} where Β() is the beta function
Median	👁 {\displaystyle \left(2^{\frac {1}{k}}-1\right)^{\frac {1}{c}}}
Mode	👁 {\displaystyle \left({\frac {c-1}{kc+1}}\right)^{\frac {1}{c}}}
Variance	👁 {\displaystyle -\mu _{1}^{2}+\mu _{2}}
Skewness	👁 {\displaystyle {\frac {2\mu _{1}^{3}-3\mu _{1}\mu _{2}+\mu _{3}}{\left(-\mu _{1}^{2}+\mu _{2}\right)^{3/2}}}}
Excess kurtosis	👁 {\displaystyle {\frac {-3\mu _{1}^{4}+6\mu _{1}^{2}\mu _{2}-4\mu _{1}\mu _{3}+\mu _{4}}{\left(-\mu _{1}^{2}+\mu _{2}\right)^{2}}}-3} where moments (see) 👁 {\displaystyle \mu _{r}=k\operatorname {\mathrm {B} } \left({\frac {ck-r}{c}},\,{\frac {c+r}{c}}\right)}
CF	👁 {\displaystyle ={\frac {c(-it)^{kc}}{\Gamma (k)}}H_{1,2}^{2,1}\!\left[(-it)^{c}\left\|{\begin{matrix}(-k,1)\\(0,1),(-kc,c)\end{matrix}}\right.\right],t\neq 0} 👁 {\displaystyle =1,t=0} where 👁 {\displaystyle \Gamma } is the Gamma function and 👁 {\displaystyle H} is the Fox H-function.^[1]

In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution^[2] is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution^[3] and is one of a number of different distributions sometimes called the "generalized log-logistic distribution".

Definitions

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Probability density function

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The Burr (Type XII) distribution has probability density function:^[4]^[5]

👁 {\displaystyle {\begin{aligned}f(x;c,k)&=ck{\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\\[6pt]f(x;c,k,\lambda )&={\frac {ck}{\lambda }}\left({\frac {x}{\lambda }}\right)^{c-1}\left[1+\left({\frac {x}{\lambda }}\right)^{c}\right]^{-k-1}\end{aligned}}}

The 👁 {\displaystyle \lambda }
parameter scales the underlying variate and is a positive real.

Cumulative distribution function

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The cumulative distribution function is:

👁 {\displaystyle F(x;c,k)=1-\left(1+x^{c}\right)^{-k}}

👁 {\displaystyle F(x;c,k,\lambda )=1-\left[1+\left({\frac {x}{\lambda }}\right)^{c}\right]^{-k}}

Applications

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It is most commonly used to model household income, see for example: Household income in the U.S. and compare to magenta graph at right.

Random variate generation

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Given a random variable 👁 {\displaystyle U}
drawn from the uniform distribution in the interval 👁 {\displaystyle \left(0,1\right)}
, the random variable

👁 {\displaystyle X=\lambda \left({\frac {1}{\sqrt[{k}]{1-U}}}-1\right)^{1/c}}

has a Burr Type XII distribution with parameters 👁 {\displaystyle c}
, 👁 {\displaystyle k}
and 👁 {\displaystyle \lambda }
. This follows from the inverse cumulative distribution function given above.

Related distributions

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When c = 1, the Burr distribution becomes the Lomax distribution.

When k = 1, the Burr distribution is a log-logistic distribution sometimes referred to as the Fisk distribution, a special case of the Champernowne distribution.^[6]^[7]

The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.^[8]

The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution

References

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^ Nadarajah, S.; Pogány, T. K.; Saxena, R. K. (2012). "On the characteristic function for Burr distributions". Statistics. 46 (3): 419–428. doi:10.1080/02331888.2010.513442. S2CID 120848446.
^ Burr, I. W. (1942). "Cumulative frequency functions". Annals of Mathematical Statistics. 13 (2): 215–232. doi:10.1214/aoms/1177731607. JSTOR 2235756.
^ Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes". Econometrica. 44 (5): 963–970. doi:10.2307/1911538. JSTOR 1911538.
^ Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN 0-521-33825-5.
^ Tadikamalla, Pandu R. (1980), "A Look at the Burr and Related Distributions", International Statistical Review, 48 (3): 337–344, doi:10.2307/1402945, JSTOR 1402945
^ C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. See Sections 7.3 "Champernowne Distribution" and 6.4.1 "Fisk Distribution."
^ Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20 (4): 591–614. doi:10.2307/1907644. JSTOR 1907644.
^ See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."

External links

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Cook, John D. (2023-02-15). "The other Burr distributions". www.johndcook.com.

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 Power function Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime 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 Hartman–Watson 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 hyperbolic Generalized logistic (logistic-beta) Generalized normal 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