Burr distribution

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

Burr Type XII

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

Definitions

Probability density function

The Burr (Type XII) distribution has probability density function:

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

The $\lambda$ parameter scales the underlying variate and is a positive real.

Cumulative distribution function

The cumulative distribution function is:

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

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

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.

Random variate generation

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

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

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

Related distributions

• 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.

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

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

Further reading

• Rodriguez, R. N. (1977). "A guide to Burr Type XII distributions". Biometrika. 64 (1): 129–134. doi:10.1093/biomet/64.1.129.

External links

• John (2023-02-16). "The other Burr distributions". www.johndcook.com.