# Burr distribution

In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution[1] is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution[2] and is one of a number of different distributions sometimes called the "generalized log-logistic distribution". 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.

Parameters Probability density function Cumulative distribution function ${\displaystyle c>0\!}$ ${\displaystyle k>0\!}$ ${\displaystyle x>0\!}$ ${\displaystyle ck{\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\!}$ ${\displaystyle 1-\left(1+x^{c}\right)^{-k}}$ ${\displaystyle \mu _{1}=k\operatorname {\mathrm {B} } (k-1/c,\,1+1/c)}$ where Β() is the beta function ${\displaystyle \left(2^{\frac {1}{k}}-1\right)^{\frac {1}{c}}}$ ${\displaystyle \left({\frac {c-1}{kc+1}}\right)^{\frac {1}{c}}}$ ${\displaystyle -\mu _{1}^{2}+\mu _{2}}$ ${\displaystyle {\frac {2\mu _{1}^{3}-3\mu _{1}\mu _{2}+\mu _{3}}{\left(-\mu _{1}^{2}+\mu _{2}\right)^{3/2}}}}$ ${\displaystyle {\frac {-3\mu _{1}^{4}+6\mu _{1}^{2}\mu _{2}-4\mu _{1}\mu _{3}+\mu _{4}}{\left(-\mu _{1}^{2}+\mu _{2}\right)^{2}}}-3}$ where moments (see) ${\displaystyle \mu _{r}=k\operatorname {\mathrm {B} } \left({\frac {ck-r}{c}},\,{\frac {c+r}{c}}\right)}$

The Burr (Type XII) distribution has probability density function:[3][4]

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

When c = 1, the Burr distribution becomes the Pareto Type II (Lomax) distribution. When k = 1, the Burr distribution is a special case of the Champernowne distribution, often referred to as the Fisk distribution.[5][6]

The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.[7]