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

The Burr distribution has probability density function:
The parameter scales the underlying variate and is a positive real.

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.

Given a random variable drawn from the uniform distribution in the interval, the random variable
has a Burr Type XII distribution with parameters, and. This follows from the inverse cumulative distribution function given above.

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, 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