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Burr distribution

Probability distribution used to model household income

Probability distribution used to model household income

name =Burr Type XII| type =density| pdf_image =[[Image:Burr pdf.svg|325px]]| cdf_image =[[Image:Burr cdf.svg|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 (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.

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:

: \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.

References

References

  1. (2012). "On the characteristic function for Burr distributions". Statistics.
  2. Burr, I. W.. (1942). "Cumulative frequency functions". [[Annals of Mathematical Statistics]].
  3. (1976). "A Function for the Size Distribution of Incomes". [[Econometrica]].
  4. Maddala, G. S.. (1996). "Limited-Dependent and Qualitative Variables in Econometrics". Cambridge University Press.
  5. Tadikamalla, Pandu R.. (1980). "A Look at the Burr and Related Distributions". International Statistical Review.
  6. C. Kleiber and S. Kotz. (2003). "Statistical Size Distributions in Economics and Actuarial Sciences". Wiley.
  7. Champernowne, D. G.. (1952). "The graduation of income distributions". [[Econometrica]].
  8. See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."
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