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".

Burr Type XII
Probability density function
Cumulative distribution function
Parameters
Support
PDF
CDF
Quantile
Mean where Β() is the beta function
Median
Mode
Variance
Skewness
Excess kurtosis where moments (see)
CF

where is the Gamma function and is the Fox H-function.[1]

Definitions

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Probability density function

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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

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The cumulative distribution function is:

 
 

Applications

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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.

Random variate generation

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Given a random variable   drawn from the uniform distribution in the interval  , the random variable

 

has a Burr Type XII distribution with parameters  ,   and  . This follows from the inverse cumulative distribution function given above.

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  • 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

References

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  1. ^ Nadarajah, S.; Pogány, T. K.; Saxena, R. K. (2012). "On the characteristic function for Burr distributions". Statistics. 46 (3): 419–428. doi:10.1080/02331888.2010.513442. S2CID 120848446.
  2. ^ Burr, I. W. (1942). "Cumulative frequency functions". Annals of Mathematical Statistics. 13 (2): 215–232. doi:10.1214/aoms/1177731607. JSTOR 2235756.
  3. ^ Singh, S.; Maddala, G. (1976). "A Function for the Size Distribution of Incomes". Econometrica. 44 (5): 963–970. doi:10.2307/1911538. JSTOR 1911538.
  4. ^ Maddala, G. S. (1996) [1983]. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press. ISBN 0-521-33825-5.
  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
  6. ^ 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."
  7. ^ Champernowne, D. G. (1952). "The graduation of income distributions". Econometrica. 20 (4): 591–614. doi:10.2307/1907644. JSTOR 1907644.
  8. ^ See Kleiber and Kotz (2003), Table 2.4, p. 51, "The Burr Distributions."

Further reading

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