Burr Distribution

The Burr Type XII distribution, often referred to simply as the Burr distribution, is a continuous probability distribution that is the most common of a family of 12 distributions proposed by Irving Burr in 1942. It is also known as the Singh-Maddala distribution and is one of a number of different distributions sometimes called the "log-logistic distribution".

The Burr distribution is defined by two shape parameters, c and k, which control the shape of the distribution. It can take on various forms depending on the values of these parameters, making it a versatile choice for modeling data.

The Burr distribution is often used to model income distributions, as it can represent both high and low incomes effectively. It can model a wide range of data, including skewed and heavy-tailed distributions. It is particularly useful in fields such as economics, finance, and reliability engineering.

Definition

The Burr distribution has two parameters, a continuous shape parameter c and a continuous shape parameter k. A scale parameter, λ may also be specified. The standard probability density function (PDF) is:

f(x;c,k)=ckxc−1(1+xc)k+1

The cumulative distribution function (CDF) is:

F(x;c,k)=1−(1+xc)−k

The Burr distribution is defined for all positive real numbers x>0. Both shape parameters c and k must be strictly positive.

Applications

The Burr distribution can be used to model a wide range of phenomena. Some examples include:

  • In economics, it is used to model income distribution.

  • In finance and insurance, it is used to model losses from insurance claims and other financial data.

  • In hydrology, it is used to model flood levels and rainfall.

  • In reliability engineering, it is used to model the lifetime of devices.

Properties

The following table lists some common properties of the Burr distribution.

Statistical Properties
PropertyValue
Meank \Beta(k-1/c),1+1/c), \quad ck > 1
VariancekΓ(k−2/c)Γ(2/c)Γ(k)−μ2,ck>2
Median(21/k−1)1/c
Mode(c−1ck+1)1/c,c>1

Relationships to Other Distributions

  • The Burr Type XII distribution is a member of the system of continuous distributions introduced by Irving W. Burr in 1942.

  • The Lomax distribution is a special case of the Burr distribution where c=1.

  • The log-logistic distribution is a special case of the Burr distribution where k=1.

The BurrDistribution Class

The Burr distribution is implemented by the BurrDistribution class. It has one constructor that takes two arguments. The first argument is the shape parameter c, and the second argument is the shape parameter k.

The following constructs a Burr distribution with shape parameters 2.5 and 3.1:

C# 
var burr = new BurrDistribution(6.8, 4.1);

The BurrDistribution class has two properties, ShapeParameter1 and ShapeParameter2, which return the shape parameters c and k of the distribution, respectively.

BurrDistribution has one static (Shared in Visual Basic) method, Sample(Random, Double, Double), which generates a random sample using a user-supplied uniform random number generator. The second and third parameters are the shape parameters of the distribution.

C# 
var random = new Pcg32();
double sample = BurrDistribution.Sample(random, 6.8, 4.1);

The above example uses the Pcg32 class to generate uniform random numbers.

References

For more information on the Burr distribution, refer to the following sources:

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

  • Singh, S. K.; Maddala, G. S. (1976). "A Function for Size Distribution of Incomes". Econometrica. 44 (5): 963–970.

  • Wikipedia, The Free Encyclopedia. Burr Type XII distribution. https://en.wikipedia.org/wiki/Burr_distribution

See Also