Burr distribution

Burr Type XII
Probability density function File:Burr pdf.svg
Cumulative distribution function File:Burr cdf.svg
Parameters	$c>0$ $k>0$
Support	$x>0$
PDF	$ck\frac{x^{c-1}}{(1+x^c{)}^{k+1}}$
CDF	$1-{(1+x^c)}^{-k}$
Quantile	$\lambda {(\frac{1}{(1-U{)}^{\frac{1}{k}}}-1)}^{\frac{1}{c}}$
Mean	${\mu }_1=k\mathrm{B}(k-1/c,1+1/c)$ where Β() is the beta function
Median	${(2^{\frac{1}{k}}-1)}^{\frac{1}{c}}$
Mode	${\left(\frac{c-1}{kc+1}\right)}^{\frac{1}{c}}$
Variance	$-{\mu }_1^2+{\mu }_2$
Skewness	$\frac{2{\mu }_1^3-3{\mu }_1{\mu }_2+{\mu }_3}{{(-{\mu }_1^2+{\mu }_2)}^{3/2}}$
Excess kurtosis	$\frac{-3{\mu }_1^4+6{\mu }_1^2{\mu }_2-4{\mu }_1{\mu }_3+{\mu }_4}{{(-{\mu }_1^2+{\mu }_2)}^2}-3$ where moments (see) ${\mu }_r=k\mathrm{B}(\frac{ck-r}{c},\frac{c+r}{c})$
CF	$=\frac{c(-it{)}^{kc}}{\Gamma (k)}{H}_{1,2}^{2,1}[(-it{)}^c|\begin{array}{c}(-k,1)\\ (0,1),(-kc,c)\end{array}],t\ne 0$ $=1,t=0$ where $\Gamma$ is the Gamma function and $H$ is the Fox H-function.

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".

Definitions

Probability density function

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

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

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

Cumulative distribution function

The cumulative distribution function is:

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

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

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 $(0,1)$, the random variable

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

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