Burr 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	$1-\left(1+x^{c}\right)^{-k}$
Mean	$k\operatorname {B} (k-1/c,\,1+1/c)$ where B() is the beta function
Median	$\left(2^{\frac {1}{k}}-1\right)^{\frac {1}{c}}$
Mode	$\left({\frac {c-1}{kc+1}}\right)^{\frac {1}{c}}$

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". It is most commonly used to model household income (See: Household income in the U.S. and compare to magenta graph at right).

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

$f(x;c,k)=ck{\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\!$

and cumulative distribution function:

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

Note when c=1, the Burr distribution becomes the Pareto Type II distribution. When k=1, the Burr distribution is a special case of the Champernowne distribution, often referred to as the Fisk distribution.

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

See also

• Dagum distribution, also known as the inverse Burr Distribution.

Further reading

• Rodriguez, R.N. (1977) "A guide to Burr Type XII distributions", Biometrika, 64, 129–134