Burr distribution
Probability density function | |
Cumulative distribution function | |
Parameters |
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Support | |
CDF | |
Mean | where Β() is the beta function |
Median | |
Mode | |
Variance | |
Skewness | |
Ex. kurtosis | where moments (see) |
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]
and cumulative distribution function:
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
- Dagum distribution, also known as the inverse Burr Distribution.
References
- ↑ Burr, I. W. (1942). "Cumulative frequency functions". Annals of Mathematical Statistics. 13 (2): 215–232. doi:10.1214/aoms/1177731607. JSTOR 2235756.
- ↑ Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes". Econometrica. 44 (5): 963–970. JSTOR 1911538.
- ↑ 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."
- ↑ Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20 (4): 591–614. doi:10.2307/1907644.
- ↑ 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". Biometrika. 64 (1): 129–134. doi:10.1093/biomet/64.1.129.