The other Burr distributions
As I mentioned in the previous post, there are 12 distributions named for Irving Burr, known as Burr Type I, Burr Type II, Burr Type III, ..., Burr Type XII. [1]
The last of these is by far the most common, and the rest are hard to find online. I did manage to find them, so I'll list them here for my future reference and for the benefit of anyone else interested in these distributions in the future.
Each distribution has a closed-form CDF because each is defined by its CDF. In all but one case, Burr Type XI, the CDF functions are invertible in closed-form. This means that except for Burr Type XI, one can easily generate random samples from each of the Burr distributions by applying the inverse CDF to a uniform random variable.
For each distribution I'll give the CDF.
Burr Type I distribution
for 0 < x < 1, which is the uniform distribution.
Burr Type II distribution
for - x
Burr Type III distribution
for 0 x
The Burr Type III distribution is also known as the Dagum distribution, and is probably the most well known of the Burr distributions after Type XII.
Burr Type IV distribution
for 0 x c.
Burr Type V distribution
for -/2 x
Burr Type VI distribution
for - x
Burr Type VII distribution
for - x
Burr Type VIII distribution
for - x
Burr Type IX distribution
for - x
Burr Type X distribution
for 0 ≤ x
Burr Type XI distribution
for 0 x
Burr Type XII distribution
for 0 ≤ x
The Burr Type XII distribution is also known as the Singh-Maddala distribution in economics.
[1] It's possible to represent the Roman numerals I through XII as single Unicode characters as described here. So if you want to get fancy, we have Burr , Burr , Burr , ..., Burr . Here , for example, is a single character, U+216B.
The post The other Burr distributions first appeared on John D. Cook.