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".

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



\begin{align} 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{align} $$

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