Generalized Pareto Distribution
The Generalized Pareto distribution is a generalization of the Pareto distribution often used in risk analysis.
The Pareto distribution has a location parameter which corresponds to the smallest possible value of the variable, a scale parameter which must be strictly greater than 0, and a shape parameter. The probability density function is:
$$f(x) = \frac{1}{b}\left(1+\frac{a(x-c)}{b}\right)^{-1-1/a}$$The Generalized Pareto distribution is implemented by the ParetoDistribution class. It has one constructor with three arguments. The first argument is the shape parameter. The second and third arguments are the scale and location parameters, respectively.
The following constructs the Generalized Pareto distribution with shape parameter -0.2, scale parameter 3, and location parameter 4.5:
var pareto = new GeneralizedParetoDistribution(-0.2, 3, 4.5);The GeneralizedParetoDistribution class has three specific properties, ShapeParameter, ScaleParameter, and LocationParameter, which return the shape, scale and location parameters of the distribution.
GeneralizedParetoDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The second, third and fourth arguments are the shape, scale and location parameters of the distribution.
var random = new Pcg32();
double sample = GeneralizedParetoDistribution.Sample(random, -0.2, 3, 4.5);The above example uses the MersenneTwister class to generate uniform random numbers.
For details of the properties and methods common to all continuous distribution classes, see the topic on continuous distributions..