Parameters Probability density function Cumulative distribution function ${\displaystyle c>0\!}$ ${\displaystyle k>0\!}$ ${\displaystyle x>0\!}$ ${\displaystyle ck{\frac {x^{c-1)){(1+x^{c})^{k+1))}\!}$ ${\displaystyle 1-\left(1+x^{c}\right)^{-k))$ ${\displaystyle \lambda \left({\frac {1}{(1-U)^{\frac {1}{k))))-1\right)^{\frac {1}{c))}$ ${\displaystyle \mu _{1}=k\operatorname {\mathrm {B} } (k-1/c,\,1+1/c)}$ where Β() is the beta function ${\displaystyle \left(2^{\frac {1}{k))-1\right)^{\frac {1}{c))}$ ${\displaystyle \left({\frac {c-1}{kc+1))\right)^{\frac {1}{c))}$ ${\displaystyle -\mu _{1}^{2}+\mu _{2))$ ${\displaystyle {\frac {2\mu _{1}^{3}-3\mu _{1}\mu _{2}+\mu _{3)){\left(-\mu _{1}^{2}+\mu _{2}\right)^{3/2))))$ ${\displaystyle {\frac {-3\mu _{1}^{4}+6\mu _{1}^{2}\mu _{2}-4\mu _{1}\mu _{3}+\mu _{4)){\left(-\mu _{1}^{2}+\mu _{2}\right)^{2))}-3}$ where moments (see) ${\displaystyle \mu _{r}=k\operatorname {\mathrm {B} } \left({\frac {ck-r}{c)),\,{\frac {c+r}{c))\right)}$ ${\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}$${\displaystyle =1,t=0}$where ${\displaystyle \Gamma }$ is the Gamma function and ${\displaystyle H}$ is the Fox H-function.[1]

In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution[2] is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution[3] 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:[4][5]

{\displaystyle {\begin{aligned}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{aligned))}

The ${\displaystyle \lambda }$ parameter scales the underlying variate and is a positive real.

### Cumulative distribution function

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

## 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 ${\displaystyle U}$ drawn from the uniform distribution in the interval ${\displaystyle \left(0,1\right)}$, the random variable

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

has a Burr Type XII distribution with parameters ${\displaystyle c}$, ${\displaystyle k}$ and ${\displaystyle \lambda }$. This follows from the inverse cumulative distribution function given above.

• The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.[8]
• The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution

## References

1. ^ Nadarajah, S.; Pogány, T. K.; Saxena, R. K. (2012). "On the characteristic function for Burr distributions". Statistics. 46 (3): 419–428. doi:10.1080/02331888.2010.513442. S2CID 120848446.
2. ^ Burr, I. W. (1942). "Cumulative frequency functions". Annals of Mathematical Statistics. 13 (2): 215–232. doi:10.1214/aoms/1177731607. JSTOR 2235756.
3. ^ Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes". Econometrica. 44 (5): 963–970. doi:10.2307/1911538. JSTOR 1911538.
4. ^ Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN 0-521-33825-5.
5. ^ Tadikamalla, Pandu R. (1980), "A Look at the Burr and Related Distributions", International Statistical Review, 48 (3): 337–344, doi:10.2307/1402945, JSTOR 1402945
6. ^ C. Kleiber and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley. See Sections 7.3 "Champernowne Distribution" and 6.4.1 "Fisk Distribution."
7. ^ Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20 (4): 591–614. doi:10.2307/1907644. JSTOR 1907644.
8. ^ See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."