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".

Definitions

Probability density function

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

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

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