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Burr distribution

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Title: Burr distribution  
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Subject: Champernowne distribution, Probability Distribution, Shape parameter, Heavy-tailed distribution, Nonparametric skew
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Burr distribution

Burr Type XII
Probability density function
Cumulative distribution function
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 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}

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


  1. ^ Burr, I. W. (1942). "Cumulative frequency functions".  
  2. ^ Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes".  
  3. ^ Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press.  
  4. ^ Tadikamalla, Pandu R. (1980), "A Look at the Burr and Related Distributions", International Statistical Review 48 (3): 337–344,  
  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".  
  7. ^ See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."

Further reading

  • Rodriguez, R. N. (1977). "A guide to Burr Type XII distributions".  
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