In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution[2] is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution[3] and is one of a number of different distributions sometimes called the "generalized log-logistic distribution".
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Burr Type XII
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where moments (see) |
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where is the Gamma function and is the Fox H-function.[1] |
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Probability density function
The Burr (Type XII) distribution has probability density function:[4][5]
The parameter scales the underlying variate and is a positive real.
Cumulative distribution function
The cumulative distribution function is:
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.
- The Burr Type XII distribution is a member of a system of continuous distributions introduced by Irving W. Burr (1942), which comprises 12 distributions.[8]
- The Dagum distribution, also known as the inverse Burr distribution, is the distribution of 1 / X, where X has the Burr distribution
Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN 0-521-33825-5.
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
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."
See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."