Burr distribution

Burr Type XII
	Probability density function
	Cumulative distribution function
Parameters	;
Support
PDF
CDF
Mean	where Β() is the beta function
Median
Mode
Variance
Skewness
Ex. kurtosis	where moments (see)
CF	; ; where is the Gamma function and 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]

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

Cumulative distribution function

The cumulative distribution function is:

F(x;c,k)=1-\left(1+x^{c}\right)^{-k}

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

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

has a Burr Type XII distribution with parameters $c$ , $k$ and $\lambda$ . 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.[6][7]

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

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.
Burr, I. W. (1942). "Cumulative frequency functions". Annals of Mathematical Statistics. 13 (2): 215–232. doi:10.1214/aoms/1177731607. JSTOR 2235756.
Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes". Econometrica. 44 (5): 963–970. doi:10.2307/1911538. JSTOR 1911538.
Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN 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
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.
See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."

External links

John (2023-02-16). "The other Burr distributions". www.johndcook.com.

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[burr_cf_cite-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.

[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."

Burr Type XII
Probability density function
Cumulative distribution function
Parameters	$c>0\!$ $k>0\!$
Support	$x>0\!$
PDF	$ck{\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\!$
CDF	$1-\left(1+x^{c}\right)^{-k}$
Mean	$\mu _{1}=k\operatorname {\mathrm {B} } (k-1/c,\,1+1/c)$ where Β() is the beta function
Median	$\left(2^{\frac {1}{k}}-1\right)^{\frac {1}{c}}$
Mode	$\left({\frac {c-1}{kc+1}}\right)^{\frac {1}{c}}$
Variance	$-\mu _{1}^{2}+\mu _{2}$
Skewness	${\frac {2\mu _{1}^{3}-3\mu _{1}\mu _{2}+\mu _{3}}{\left(-\mu _{1}^{2}+\mu _{2}\right)^{3/2}}}$
Ex. kurtosis	${\frac {-3\mu _{1}^{4}+6\mu _{1}^{2}\mu _{2}-4\mu _{1}\mu _{3}+\mu _{4}}{\left(-\mu _{1}^{2}+\mu _{2}\right)^{2}}}-3$ where moments (see) $\mu _{r}=k\operatorname {\mathrm {B} } \left({\frac {ck-r}{c}},\,{\frac {c+r}{c}}\right)$
CF	$={\frac {c(-it)^{kc}}{\Gamma (k)}}H_{1,2}^{2,1}\!\left[(-it)^{c}\left\|{\begin{matrix}(-k,1)\\(0,1),(-kc,c)\end{matrix}}\right.\right],t\neq 0$ $=1,t=0$ where $\Gamma$ is the Gamma function and $H$ is the Fox H-function.[1]