Burr distribution

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Burr Type XII
Probability density function
Burr pdf.svg
Cumulative distribution function
Burr cdf.svg
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{\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 2\mu_1^3-3\mu_1\mu_2+\mu_3
Ex. kurtosis -3\mu_1^4+6\mu_1^2\mu_2-4\mu_1\mu_3+\mu_4 where moments (see) \mu_r =k\operatorname{\Beta}\left(\frac{ck-r}{c},\, \frac{c+r}{c}\right)

In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution[1] is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution[2] 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: Household income in the U.S. and compare to magenta graph at right).

The Burr (Type XII) distribution has probability density function:[3][4]

f(x;c,k) = ck\frac{x^{c-1}}{(1+x^c)^{k+1}}\!

and cumulative distribution function:

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

Note when c=1, the Burr distribution becomes the Pareto Type II distribution. When k=1, the Burr distribution is a special case of the Champernowne distribution, often referred to as the Fisk distribution.[5][6]

The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.[7]

See also[edit]


  1. ^ Burr, I. W. (1942). "Cumulative frequency functions". Annals of Mathematical Statistics 13 (2): 215–232. doi:10.1214/aoms/1177731607. JSTOR 2235756. 
  2. ^ Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes". Econometrica 44 (5): 963–970. JSTOR 1911538. 
  3. ^ Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN 0-521-33825-5. 
  4. ^ 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 
  5. ^ 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."
  6. ^ Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica 20 (4): 591–614. doi:10.2307/1907644. 
  7. ^ See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."

Further reading[edit]