Burr distribution

Burr Type XII

Probability density function
Cumulative distribution function
Parameters	[math]\displaystyle{ c \gt 0\! }[/math] [math]\displaystyle{ k \gt 0\! }[/math]
Support	[math]\displaystyle{ x \gt 0\! }[/math]
PDF	[math]\displaystyle{ ck\frac{x^{c-1}}{(1+x^c)^{k+1}}\! }[/math]
CDF	[math]\displaystyle{ 1-\left(1+x^c\right)^{-k} }[/math]
Mean	[math]\displaystyle{ \mu_1=k\operatorname{\Beta}(k-1/c,\, 1+1/c) }[/math] where Β() is the beta function
Median	[math]\displaystyle{ \left(2^{\frac{1}{k}}-1\right)^\frac{1}{c} }[/math]
Mode	[math]\displaystyle{ \left(\frac{c-1}{kc+1}\right)^\frac{1}{c} }[/math]
Variance	[math]\displaystyle{ -\mu_1^2+\mu_2 }[/math]
Skewness	[math]\displaystyle{ \frac{ 2\mu _{1}^{3}-3\mu _{1}\mu _{2}+\mu _{3}}{\left( -\mu _{1}^{2}+\mu _{2}\right)^{3/2}} }[/math]
Kurtosis	[math]\displaystyle{ \frac{-3\mu _{1}^{4}+6\mu _{1}^{2}\mu _{2}-4\mu _{1}\mu _{3}+\mu _{4}}{\left( -\mu _{1}^{2}+\mu _{2}\right)^{2}}-3 }[/math] where moments (see) [math]\displaystyle{ \mu_r =k\operatorname{\Beta}\left(\frac{ck-r}{c},\, \frac{c+r}{c}\right) }[/math]
CF	[math]\displaystyle{ = \frac{c(-it)^{kc}}{\Gamma(k)}H_{1,2}^{2,1}\!\left[(-it)^c\left| \begin{matrix} (-k, 1)\\(0, 1),(-kc,c)\end{matrix}\right. \right], t\neq 0 }[/math] [math]\displaystyle{ = 1, t = 0 }[/math] where [math]\displaystyle{ \Gamma }[/math] is the Gamma function and [math]\displaystyle{ H }[/math] is the Fox H-function.

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:

[math]\displaystyle{ \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} }[/math]

The [math]\displaystyle{ \lambda }[/math] parameter scales the underlying variate and is a positive real.

Cumulative distribution function

The cumulative distribution function is:

[math]\displaystyle{ F(x;c,k) = 1-\left(1+x^c\right)^{-k} }[/math]

[math]\displaystyle{ F(x;c,k,\lambda) = 1 - \left[1 + \left(\frac{x}{\lambda}\right)^c \right]^{-k} }[/math]

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 [math]\displaystyle{ U }[/math] drawn from the uniform distribution in the interval [math]\displaystyle{ \left(0, 1\right) }[/math], the random variable

[math]\displaystyle{ X=\lambda \left (\frac{1}{\sqrt[k]{1-U}}-1 \right )^{1/c} }[/math]

has a Burr Type XII distribution with parameters [math]\displaystyle{ c }[/math],[math]\displaystyle{ k }[/math] and [math]\displaystyle{ \lambda }[/math]. 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

Further reading

• Rodriguez, R. N. (1977). "A guide to Burr Type XII distributions". Biometrika 64 (1): 129–134. doi:10.1093/biomet/64.1.129. http://www.lib.ncsu.edu/resolver/1840.4/3481. 

External links

• John (2023-02-16). "The other Burr distributions" (in en-US). https://www.johndcook.com/blog/2023/02/15/all-burr-distributions/.