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:
The parameter scales the underlying variate and is a positive real.
Cumulative distribution function
The cumulative distribution function is:
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 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.
Related distributions
When c = 1, the Burr distribution becomes the Pareto Type II (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