Burr Type XII
Probability density function ![](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Burr_pdf.svg/325px-Burr_pdf.svg.png) |
Cumulative distribution function ![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Burr_cdf.svg/325px-Burr_cdf.svg.png) |
Parameters |
![{\displaystyle k>0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/125619fdc1f9e278be47ee4df1668c733aff3085) |
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Support |
![{\displaystyle x>0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56eb4a141d33a74ab2c6059b071f9f8e7897848c) |
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PDF |
![{\displaystyle ck{\frac {x^{c-1)){(1+x^{c})^{k+1))}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d40e1334f14aeedab3593cd56253ea07b2c907c5) |
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CDF |
![{\displaystyle 1-\left(1+x^{c}\right)^{-k))](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b174f815aefb386d33176bbd371bf4067735b0) |
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Quantile |
![{\displaystyle \lambda \left({\frac {1}{(1-U)^{\frac {1}{k))))-1\right)^{\frac {1}{c))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0e9da15b9debea1ab8c6285815f5c7150a58c7) |
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Mean |
where Β() is the beta function |
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Median |
![{\displaystyle \left(2^{\frac {1}{k))-1\right)^{\frac {1}{c))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3980ca09a27e43045e8935c84660a5bbb61fe4) |
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Mode |
![{\displaystyle \left({\frac {c-1}{kc+1))\right)^{\frac {1}{c))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f7068c91659c167f7cbe8ee3ed8ea0b8006ab11) |
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Variance |
![{\displaystyle -\mu _{1}^{2}+\mu _{2))](https://wikimedia.org/api/rest_v1/media/math/render/svg/a62b9524f34a2b434b271f76c6ef52d555887994) |
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Skewness |
![{\displaystyle {\frac {2\mu _{1}^{3}-3\mu _{1}\mu _{2}+\mu _{3)){\left(-\mu _{1}^{2}+\mu _{2}\right)^{3/2))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/9184628a057029958060309cec996609f6c736c1) |
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Excess kurtosis |
where moments (see) ![{\displaystyle \mu _{r}=k\operatorname {\mathrm {B} } \left({\frac {ck-r}{c)),\,{\frac {c+r}{c))\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a633f103124ac538eefca023c56a491e296ddae) |
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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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d57864ffe50078a34292d9f949f09e0f47ab2b3)
![{\displaystyle =1,t=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b45059037a65b9491aa80e8df61a642a4e8ac3a) where is the Gamma function and is the Fox H-function.[1] |
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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
Probability density function
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c758cb4945e4a43a37fb66ad628b53ca8de40c)
The
parameter scales the underlying variate and is a positive real.
Cumulative distribution function
The cumulative distribution function is:
![{\displaystyle F(x;c,k)=1-\left(1+x^{c}\right)^{-k))](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb035db539f43b515d817bc0400fde1ddfd683d3)
![{\displaystyle F(x;c,k,\lambda )=1-\left[1+\left({\frac {x}{\lambda ))\right)^{c}\right]^{-k))](https://wikimedia.org/api/rest_v1/media/math/render/svg/00238e8a5ecdca4563db2344bfca37b93baee67f)
Applications
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.