Английская Википедия:Burr distribution

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Шаблон:Probability distribution{(1+x^c)^{k+1}}\!</math>|

 cdf        =<math>1-\left(1+x^c\right)^{-k}</math>|
 mean       =<math>\mu_1=k\operatorname{\Beta}(k-1/c,\, 1+1/c)</math> where Β() is the beta function|
 median     =<math>\left(2^{\frac{1}{k}}-1\right)^\frac{1}{c}</math>|
 mode       =<math>\left(\frac{c-1}{kc+1}\right)^\frac{1}{c}</math>|
 variance   =<math>-\mu_1^2+\mu_2</math>|
 skewness   =<math>\frac{ 2\mu _{1}^{3}-3\mu _{1}\mu _{2}+\mu _{3}}{\left( -\mu _{1}^{2}+\mu _{2}\right)^{3/2}}</math>|
 kurtosis   =<math>\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</math> where moments (see) <math>\mu_r =k\operatorname{\Beta}\left(\frac{ck-r}{c},\, \frac{c+r}{c}\right)</math>|
 entropy    =|
 mgf        =|
 char       = <math>= \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</math>
<math>= 1, t = 0</math>
where <math>\Gamma</math> is the Gamma function and <math>H</math> is the Fox H-function. }}

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:

<math>

\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} </math>

The <math>\lambda</math> parameter scales the underlying variate and is a positive real.

Cumulative distribution function

The cumulative distribution function is:

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

<math>F(x;c,k,\lambda) = 1 - \left[1 + \left(\frac{x}{\lambda}\right)^c \right]^{-k}</math>

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 <math>U</math> drawn from the uniform distribution in the interval <math>\left(0, 1\right)</math>, the random variable

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

has a Burr Type XII distribution with parameters <math>c</math>, <math>k</math> and <math>\lambda</math>. 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.

• The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.

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

Further reading

• Шаблон:Cite journal

External links

• Шаблон:Cite web