# Burr Distribution

The Burr distribution (also called the Burr Type XII distribution or Singh–Maddala distribution) is a unimodal family of distributions with a wide variety of shapes. “Unimodal” means that the distribution has one peak. This distribution is used to model a wide variety of phenomena including crop prices, household income, option market price distributions, risk (insurance) and time to destination. It is particularly useful for modeling histograms.

The Burr distribution is very similar (and is, in some cases, the same as) many other distributions such as:

• Compounded Weibull with a gamma distribution as its shape parameter.
• Gamma distribution,
• J-shaped beta distribution,
• Loglogistic distribution,
• Lognormal distribution,
• Normal distribution.

## Types I to XII

In 1941, Burr introduced twelve cumulative distribution functions (CDFs) that could be fit to real life data. However, the Burr Type XII family was the only one he originally studied in depth; the others were studied in depth at later dates. Thus, the term “Burr distribution” usually refers to type XII.

• Burr Type I family = the uniform distribution.
• Burr Type II distribution = the generalized logistic distribution.
• Burr Type III (also called the inverse Burr distribution or Dagum type distribution) is (along with type XII) commonly used for statistical modeling. To obtain this distribution from the PDF of the Burr type II: replace “X” in the PDF with “ln(x)” (Johnson et. al).
• Burr Type IV: When the location parameter (l) = 0 and scale parameter (s) = 1, it becomes the standard Burr type VI distribution.
• Burr Type X: the same as the generalized Rayleigh distribution.
• A five-parameter distribution, the beta Burr XII, is useful for modeling lifetime data. It is called the Singh–Maddala distribution in economics.

Most forms of this distribution have little research associated with them. For example, Feroze et. al (2013) say about the Type V that “Many properties of the parameters of the distribution under different estimation procedures are still to be revealed.” Thus, the CDFs for these types can be challenging to track down. Credit to John D. Cook for finding the CDFs for all 12 online:

## Burr distribution definitions and properties

The Burr distribution (Type XII) is the most common Type you’ll come across, thus I’ll dive in deeper here to its properties. It is defined by the following parameters:

• c and k: shape parameters. For Burr Type XII, these are both positive. Type III has a negative c parameter.
• α: scale parameter.
• γ: continuous location parameter.

When the fourth parameter, γ, equals zero, it gives a three parameter (c,k,α) distribution.

A given set of data can be matched to a Burr distribution by matching the mean, kurtosis, skewness and variance of the data set.

The probability density function (PDF) for the Burr VI distribution is:

## References

Top image: I, Selket, CC BY-SA 3.0 http://creativecommons.org/licenses/by-sa/3.0/, via Wikimedia Commons
Burr, I. W. (1942). “Cumulative frequency functions”. Annals of Mathematical Statistics 13 (2): 215–232.

Cook, J. The Other Burr Distributions: https://www.johndcook.com/blog/2023/02/15/all-burr-distributions/
Feroze, N. & Aslam, M. (2013) “Maximum Likelihood Estimation of Burr Type V Distribution under Left Censored Samples.” WSEAS Transactions on Mathematics. Retrieved October 7, 2016 from here.
Johnson, N.L. et. al (1995). “Continuous Univariate Distributions”. Vol. 2, John Wiley & Sons, New York, NY, USA, 2nd edition.
Tadikamalla, Pandu R. (1980), “A Look at the Burr and Related Distributions”, International Statistical Review 48 (3): 337–344