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 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. See Sections 7.3 "Champernowne Distribution" and 6.4.1 "Fisk 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.See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."
The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution
References
Further reading
External links