Burr distribution – Wikipedia

Burr Type XII

Probability density function
Cumulative distribution function
Parameters	${displaystyle c>0!}$ ${displaystyle k>0!}$ Support			${displaystyle x>0!}$ PDF			${displaystyle ck{frac {x^{c-1}}{(1+x^{c})^{k+1}}}!}$
CDF	${displaystyle 1-left(1+x^{c}right)^{-k}}$
Mean	${displaystyle mu _{1}=koperatorname {mathrm {B} } (k-1/c,,1+1/c)}$ where Β() is the beta function
Median	${displaystyle left(2^{frac {1}{k}}-1right)^{frac {1}{c}}}$
Mode	${displaystyle left({frac {c-1}{kc+1}}right)^{frac {1}{c}}}$
Variance	${displaystyle -mu _{1}^{2}+mu _{2}}$
Skewness	${displaystyle {frac {2mu _{1}^{3}-3mu _{1}mu _{2}+mu _{3}}{left(-mu _{1}^{2}+mu _{2}right)^{3/2}}}}$
Ex. kurtosis	${displaystyle {frac {-3mu _{1}^{4}+6mu _{1}^{2}mu _{2}-4mu _{1}mu _{3}+mu _{4}}{left(-mu _{1}^{2}+mu _{2}right)^{2}}}-3}$ where moments (see) ${displaystyle mu _{r}=koperatorname {mathrm {B} } left({frac {ck-r}{c}},,{frac {c+r}{c}}right)}$
CF	${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],tneq 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”. 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.

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}}}$

and cumulative distribution function:

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

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

Related distributions[edit]

• 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[edit]

Further reading[edit]

External links[edit]