Generalized Pareto Distribution

The Generalized Pareto distribution is a generalization of the Pareto distribution often used in risk analysis.

The Generalized Pareto distribution is characterized by its probability density function (PDF):

And its cumulative distribution function (CDF):

The distribution has three parameters:

• $a$ - the shape parameter (must be non-zero)

• $b$ - the scale parameter (must be positive)

• $c$ - the location parameter

The support of the distribution depends on the shape parameter:

• When $a\ge 0$, the distribution extends from $c$ to infinity

• When $a<0$, the distribution is bounded above, with support from $c$ to $c-b/a$

• Extreme value analysis in risk management and insurance

• Environmental modeling of extreme events like floods and rainfall

• Financial risk assessment for large market movements

• Reliability engineering for modeling component lifetimes

• Network traffic analysis for heavy-tailed phenomena

The distribution has several notable properties:

• Heavy-tailed behavior when $a>0$

• Upper bounded when $a<0$

• Memoryless property in the $a=0$ limit case

Relationships to Other Distributions

• The Generalized Pareto distribution is a generalization of the Pareto distribution.

• When $a=0$, it reduces to the exponential distribution.

• When $a=-1$, it becomes the continuous uniform distribution on $[c,c+b]$.

• It is the limit distribution for excesses over high thresholds in extreme value theory

• When $a=0$, it reduces to the exponential distribution.

• When $a=-1$, it becomes the continuous uniform distribution on $[c,c+b]$.

• It is the limit distribution for excesses over high thresholds in extreme value theory

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);

Dim pareto = New GeneralizedParetoDistribution(-0.2, 3, 4.5)

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

let pareto = GeneralizedParetoDistribution(-0.2, 3.0, 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);

Dim random = New Pcg32()
Dim sample = GeneralizedParetoDistribution.Sample(random, -0.2, 3, 4.5)

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

let random = Pcg32()
let sample = GeneralizedParetoDistribution.Sample(random, -0.2, 3.0, 4.5)

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

For more information on the Generalized Pareto distribution, see the following sources:

• Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 20. Wiley Series in Probability and Statistics.

• Smith, R.L. (1987). "Estimating Tails of Probability Distributions". Annals of Statistics.

• Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). "Modelling Extremal Events for Insurance and Finance". Springer.

Other Resources