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:34
f ( x ; c , k ) = c k x c − 1 ( 1 + x c ) k + 1 f ( x ; c , k , λ ) = c k λ ( x λ ) c − 1 [ 1 + ( x λ ) c ] − k − 1 {\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 λ {\displaystyle \lambda } parameter scales the underlying variate and is a positive real.
Cumulative distribution function
The cumulative distribution function is:
F ( x ; c , k ) = 1 − ( 1 + x c ) − k {\displaystyle F(x;c,k)=1-\left(1+x^{c}\right)^{-k}} F ( x ; c , k , λ ) = 1 − [ 1 + ( x λ ) c ] − 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 U {\displaystyle U} drawn from the uniform distribution in the interval ( 0 , 1 ) {\displaystyle \left(0,1\right)} , the random variable
X = λ ( 1 1 − U k − 1 ) 1 / c {\displaystyle X=\lambda \left({\frac {1}{\sqrt[{k}]{1-U}}}-1\right)^{1/c}}has a Burr Type XII distribution with parameters c {\displaystyle c} , k {\displaystyle k} and λ {\displaystyle \lambda } . 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.56
- The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.7
- 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.
External links
- John (2023-02-16). "The other Burr distributions". www.johndcook.com.
References
Burr, I. W. (1942). "Cumulative frequency functions". Annals of Mathematical Statistics. 13 (2): 215–232. doi:10.1214/aoms/1177731607. JSTOR 2235756. https://doi.org/10.1214%2Faoms%2F1177731607 ↩
Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes". Econometrica. 44 (5): 963–970. doi:10.2307/1911538. JSTOR 1911538. /wiki/Econometrica ↩
Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN 0-521-33825-5. 0-521-33825-5 ↩
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 /wiki/Doi_(identifier) ↩
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." ↩
Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20 (4): 591–614. doi:10.2307/1907644. JSTOR 1907644. /wiki/Econometrica ↩
See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions." ↩