Burr distribution

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 for example: Household income in the U.S. and compare to magenta graph at right.

Burr Type XII
Probability density function
Cumulative distribution function
Mean where Β() is the beta function
Ex. kurtosis where moments (see)

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

and cumulative distribution function:

When c = 1, the Burr distribution becomes the Pareto Type II (Lomax) 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". 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. doi:10.2307/1911538. 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. JSTOR 1907644.
  7. See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."

Further reading

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