Transformed Beta Distribution

The Transformed Beta distribution, also known as the Generalized Beta distribution, is a versatile distribution often used in loss models. It is characterized by three shape parameters and a scale parameter, allowing it to model a wide range of behaviors.

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

The distribution is defined for values of $x$ between 0 and $s$ inclusive, where all parameters ($a$, $b$, $c$, and $s$) must be strictly positive.

• Insurance mathematics uses it for modeling claim sizes and losses.

• Reliability engineering employs it for lifetime modeling.

• Risk analysis uses it for modeling uncertain quantities.

Notable properties include:

• The distribution is unimodal when $ac>1$ and $b>1$.

• The distribution is bounded on $[0,s]$.

• The shape is highly flexible due to its three shape parameters.

The Transformed Beta distribution defines a rich family of distributions. Special cases include:

• When $c=1$, it reduces to the scaled Beta distribution with scale parameter $s$.

• When $a=b=1$, it becomes a Power Function distribution.

• When $c=1$, it yields the Burr distribution.

• When $b=1$, it yields the Generalized Pareto distribution.

• When $a=1$, it yields the Inverse Burr distribution.

• When $a=b=1$, it yields the Inverse Pareto distribution.

• When $a=1$ and $b=c$, it yields the Inverse Paralogistic distribution.

• When $a=1$ and $c=1$, it yields the Loglogistic distribution.

• When $c=1$ and $a=b$, it yields the Paralogistic distribution.

• When $b=c=1$, it yields the Pareto distribution.

The distribution is also related to the Kumaraswamy distribution through parameter transformation.

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 parameter. All parameters must be strictly greater than 0.

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

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

No code example is currently available or this language may not be supported.

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 Pcg32();
double sample = TransformedBetaDistribution.Sample(random, 2, 3, 4, 5);

Dim random = New Pcg32()
Dim sample = TransformedBetaDistribution.Sample(random, 2, 3, 4, 5)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = TransformedBetaDistribution.Sample(random, 2.0, 3.0, 4.0, 5.0)

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

• Venter, G. (1994). "Transformed beta and gamma distributions and aggregate losses"

  Proceedings of the Casualty Actuarial Society 70 (133), 289-308

• Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

• Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

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

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