Probability Distributions > Burr Distribution

## What is the Burr Distribution?

The Burr distribution (sometimes called the Burr Type XII distribution or Singh–Maddala distribution) is a unimodal family of distributions with a wide variety of shapes. This distribution is used to model a wide variety of phenomena including crop prices, household income, option market price distributions, risk (insurance) and travel time. It is particularly useful for modeling histograms. Although other forms exist (the type III is also very common), the term “Burr distribution”*usually*refers to type XII.

The Burr distribution Type XII 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 pdf for the Burr XII distribution is:

The cdf is:

## Similarity to Other Distributions

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

- A compounded Weibull with a gamma distribution as its shape parameter.
- The gamma distribution,
- The J-shaped beta distribution,
- The loglogistic distribution,
- The lognormal distribution,
- The normal distribution.

## Other Types of Burr Distributions

In 1941, Burr introduced twelve cumulative distribution functions 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.

- The
**Burr I family**is the same as the uniform distribution. - The
**Burr Type II distribution**is the same as the generalized logistic distribution. - The
**Burr Type III**(also called the inverse Burr distribution or Dagum type distribution) is (along with type XII) commonly used for statistical modeling. This simple distribution is can be obtained from the PDF of the Burr type II: replace “X” in the PDF with “ln(x)” (Johnson et. al). **Burr Type IV**: usually defined by the CDF^{(NIST)}F(x;r,c) = [1 + (c-x)/x)**(1/c)]**(-r) 0 < x < c; c, r > 0 -INF < x < INF; r, k, s > 0″>. 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.

Other forms of this distribution have very 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.”

**References:**

Burr, I. W. (1942). “Cumulative frequency functions”. Annals of Mathematical Statistics 13 (2): 215–232.

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