Burr distribution

{{Short description|Probability distribution used to model household income}}

{{Probability distribution |

name =Burr Type XII|

type =density|

pdf_image =325px|

cdf_image =325px|

parameters = $c > 0\!$
$k > 0\!$ |

support = $x > 0\!$ |

pdf = $ck\frac{x^{c-1}}{(1+x^c)^{k+1}}\!$ |

cdf = $1-\left(1+x^c\right)^{-k}$ |

quantile = $\lambda \left (\frac{1}{(1-U)^{\frac{1}{k}}}-1 \right )^\frac{1}{c}$ |

mean = $\mu_1=k\operatorname{\Beta}(k-1/c,\, 1+1/c)$ 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 = $\frac{ 2\mu _{1}^{3}-3\mu _{1}\mu _{2}+\mu _{3}}{\left( -\mu _{1}^{2}+\mu _{2}\right)^{3/2}}$ |

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 ([http://www.causascientia.org/math_stat/Dists/Compendium.pdf see]) $\mu_r =k\operatorname{\Beta}\left(\frac{ck-r}{c},\, \frac{c+r}{c}\right)$ |

entropy =|

mgf =|

char = $= \frac{c(-it)^{kc}}{\Gamma(k)}H_{1,2}^{2,1}\!\left[(-it)^c\left| \begin{matrix}
(-k, 1)\\(0, 1),(-kc,c)\end{matrix}\right. \right], t\neq 0$
$= 1, t = 0$
where $\Gamma$ is the Gamma function and $H$ is the Fox H-function.{{cite journal |last2=Pogány |first2=T. K. |last1=Nadarajah |first1=S. |last3=Saxena|first3=R. K.|year=2012 |title= On the characteristic function for Burr distributions |journal=Statistics |volume=46 |issue=3 |pages=419–428 |doi=10.1080/02331888.2010.513442 |s2cid=120848446 }}

}}

In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution{{cite journal |last=Burr |first=I. W. |year=1942 |title=Cumulative frequency functions |journal=Annals of Mathematical Statistics |volume=13 |issue=2 |pages=215–232 |jstor=2235756 |doi=10.1214/aoms/1177731607|doi-access=free }} is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution{{cite journal |last2=Maddala |first2=G. |last1=Singh |first1=S. |year=1976 |title=A Function for the Size Distribution of Incomes |journal=Econometrica |volume=44 |issue=5 |pages=963–970 |jstor=1911538 |doi=10.2307/1911538 }} 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:{{cite book |last=Maddala |first=G. S. |orig-year=1983 |year=1996 |title=Limited-Dependent and Qualitative Variables in Econometrics |publisher=Cambridge University Press |isbn=0-521-33825-5 }}{{Citation|doi=10.2307/1402945|title=A Look at the Burr and Related Distributions| first=Pandu R.| last=Tadikamalla| journal=International Statistical Review| volume=48| year=1980| pages=337–344|issue=3|jstor=1402945}}

: $\begin{align}
f(x;c,k) & = ck\frac{x^{c-1}}{(1+x^c)^{k+1}} \\[6pt]
f(x;c,k,\lambda) & = \frac{ck}{\lambda} \left( \frac{x}{\lambda} \right)^{c-1} \left[1 + \left(\frac{x}{\lambda}\right)^c\right]^{-k-1}
\end{align}$

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 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.{{cite book|author=C. Kleiber and S. Kotz|title=Statistical Size Distributions in Economics and Actuarial Sciences|publisher=Wiley| location=New York|year = 2003}} See Sections 7.3 "Champernowne Distribution" and 6.4.1 "Fisk Distribution."{{cite journal|last=Champernowne |first=D. G.| journal=Econometrica | title= The graduation of income distributions | year=1952 | volume=20 |issue=4 | pages = 591–614 | doi=10.2307/1907644|jstor=1907644}}

The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."

The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution

References

External links

{{Cite web |last=Cook | first=John D. |date=2023-02-15 |title=The other Burr distributions |url=https://www.johndcook.com/blog/2023/02/15/all-burr-distributions/ |website=www.johndcook.com |language=en-US}}

Category:Continuous distributions

Category:Systems of probability distributions