The Transformed Beta Distribution

The Transformed Beta distribution is often used in loss models.

The transformed Beta distribution has three shape parameters, a, b, and c, and a scale parameter s. Its probability density function (PDF) is:

The transformed beta distribution defines a rich family of distributions. Special values of the parameters result in a variety of well-known distributions:

• The Burr distribution when c = 1.

• The Generalized Pareto distribution when b = 1.

• The Inverse Burr distribution when a = 1.

• The Inverse Pareto distribution when a = 1 and b = 1.

• The Inverse Paralogistic distribution when a = 1 and b = c.

• The Loglogistic distribution when a = 1 and c = 1.

• The Paralogistic distribution when c = 1 and a = b.

• The Pareto distribution when b = 1 and c = 1.

The transformed beta distribution is implemented by the TransformedBetaDistribution class. It has one constructor that takes four arguments: the three shape parameters and the scale parameters. All parameters must be strictly greater than 0.

var trbeta1 = new TransformedBetaDistribution(2, 3, 4, 5);

Dim trbeta1 = New TransformedBetaDistribution(2, 3, 4, 5)

let trbeta1 = TransformedBetaDistribution(2.0, 3.0, 4.0, 5.0)

The TransformedBetaDistribution class has four specific properties that correspond to the parameters of the distribution. The ShapeParameter1, ShapeParameter2, and ShapeParameter3 properties return the shape parameters. The ScaleParameter property returns the scale parameter.

TransformedBetaDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator.

var random = new MersenneTwister();
double sample = TransformedBetaDistribution.Sample(random, 2, 3, 4, 5);

Dim random = New MersenneTwister()
Dim sample = TransformedBetaDistribution.Sample(random, 2, 3, 4, 5)

let random = MersenneTwister()
let sample = TransformedBetaDistribution.Sample(random, 2.0, 3.0, 4.0, 5.0)

For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions..