GeneralizedParetoDistribution Class

Generalized Pareto distribution (three parameters).

Inheritance Hierarchy

SystemObject
  Accord.Statistics.DistributionsDistributionBase
    Accord.Statistics.Distributions.UnivariateUnivariateContinuousDistribution
      Accord.Statistics.Distributions.UnivariateGeneralizedParetoDistribution

Namespace: Accord.Statistics.Distributions.Univariate
Assembly: Accord.Statistics (in Accord.Statistics.dll) Version: 3.8.0

Syntax

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[SerializableAttribute]
public class GeneralizedParetoDistribution : UnivariateContinuousDistribution, 
	IFormattable

<SerializableAttribute>
Public Class GeneralizedParetoDistribution
	Inherits UnivariateContinuousDistribution
	Implements IFormattable

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The GeneralizedParetoDistribution type exposes the following members.

Constructors

	Name	Description
	GeneralizedParetoDistribution	Initializes a new instance of the GeneralizedParetoDistribution class.
	GeneralizedParetoDistribution(Double, Double, Double)	Initializes a new instance of the GeneralizedParetoDistribution class.

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Properties

	Name	Description
	Entropy	Gets the entropy for this distribution. (Overrides UnivariateContinuousDistributionEntropy.)
	Location	Gets the location parameter μ (mu).
	Mean	Gets the mean for this distribution. (Overrides UnivariateContinuousDistributionMean.)
	Median	Gets the median for this distribution. (Overrides UnivariateContinuousDistributionMedian.)
	Mode	Gets the mode for this distribution. (Inherited from UnivariateContinuousDistribution.)
	Quartiles	Gets the Quartiles for this distribution. (Inherited from UnivariateContinuousDistribution.)
	Scale	Gets the scale parameter σ (sigma).
	Shape	Gets shape parameter ξ (Xi).
	StandardDeviation	Gets the Standard Deviation (the square root of the variance) for the current distribution. (Inherited from UnivariateContinuousDistribution.)
	Support	Gets the support interval for this distribution. (Overrides UnivariateContinuousDistributionSupport.)
	Variance	Gets the variance for this distribution. (Overrides UnivariateContinuousDistributionVariance.)

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Methods

	Name	Description
	Clone	Creates a new object that is a copy of the current instance. (Overrides DistributionBaseClone.)
	ComplementaryDistributionFunction	Gets the complementary cumulative distribution function (ccdf) for this distribution evaluated at point x. This function is also known as the Survival function. (Inherited from UnivariateContinuousDistribution.)
	CumulativeHazardFunction	Gets the cumulative hazard function for this distribution evaluated at point x. (Inherited from UnivariateContinuousDistribution.)
	DistributionFunction(Double)	Gets the cumulative distribution function (cdf) for this distribution evaluated at point x. (Inherited from UnivariateContinuousDistribution.)
	DistributionFunction(Double, Double)	Gets the cumulative distribution function (cdf) for this distribution in the semi-closed interval (a; b] given as P(a < X ≤ b). (Inherited from UnivariateContinuousDistribution.)
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	Fit(Double)	Fits the underlying distribution to a given set of observations. (Inherited from UnivariateContinuousDistribution.)
	Fit(Double, IFittingOptions)	Fits the underlying distribution to a given set of observations. (Inherited from UnivariateContinuousDistribution.)
	Fit(Double, Double)	Fits the underlying distribution to a given set of observations. (Inherited from UnivariateContinuousDistribution.)
	Fit(Double, Int32)	Fits the underlying distribution to a given set of observations. (Inherited from UnivariateContinuousDistribution.)
	Fit(Double, Double, IFittingOptions)	Fits the underlying distribution to a given set of observations. (Inherited from UnivariateContinuousDistribution.)
	Fit(Double, Int32, IFittingOptions)	Fits the underlying distribution to a given set of observations. (Inherited from UnivariateContinuousDistribution.)
	Generate	Generates a random observation from the current distribution. (Inherited from UnivariateContinuousDistribution.)
	Generate(Random)	Generates a random observation from the current distribution. (Overrides UnivariateContinuousDistributionGenerate(Random).)
	Generate(Int32)	Generates a random vector of observations from the current distribution. (Inherited from UnivariateContinuousDistribution.)
	Generate(Int32, Double)	Generates a random vector of observations from the current distribution. (Inherited from UnivariateContinuousDistribution.)
	Generate(Int32, Random)	Generates a random vector of observations from the current distribution. (Inherited from UnivariateContinuousDistribution.)
	Generate(Int32, Double, Random)	Generates a random vector of observations from the current distribution. (Overrides UnivariateContinuousDistributionGenerate(Int32, Double, Random).)
	GetHashCode	Serves as the default hash function. (Inherited from Object.)
	GetRange	Gets the distribution range within a given percentile. (Inherited from UnivariateContinuousDistribution.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	HazardFunction	Gets the hazard function, also known as the failure rate or the conditional failure density function for this distribution evaluated at point x. (Inherited from UnivariateContinuousDistribution.)
	InnerComplementaryDistributionFunction	Gets the complementary cumulative distribution function (ccdf) for this distribution evaluated at point x. This function is also known as the Survival function. (Inherited from UnivariateContinuousDistribution.)
	InnerDistributionFunction	Gets the cumulative distribution function (cdf) for this distribution evaluated at point x. (Overrides UnivariateContinuousDistributionInnerDistributionFunction(Double).)
	InnerInverseDistributionFunction	Gets the inverse of the cumulative distribution function (icdf) for this distribution evaluated at probability p. This function is also known as the Quantile function. (Overrides UnivariateContinuousDistributionInnerInverseDistributionFunction(Double).)
	InnerLogProbabilityDensityFunction	Gets the log-probability density function (pdf) for this distribution evaluated at point x. (Inherited from UnivariateContinuousDistribution.)
	InnerProbabilityDensityFunction	Gets the probability density function (pdf) for this distribution evaluated at point x. (Overrides UnivariateContinuousDistributionInnerProbabilityDensityFunction(Double).)
	InverseDistributionFunction	Gets the inverse of the cumulative distribution function (icdf) for this distribution evaluated at probability p. This function is also known as the Quantile function. (Inherited from UnivariateContinuousDistribution.)
	LogCumulativeHazardFunction	Gets the log of the cumulative hazard function for this distribution evaluated at point x. (Inherited from UnivariateContinuousDistribution.)
	LogProbabilityDensityFunction	Gets the log-probability density function (pdf) for this distribution evaluated at point x. (Inherited from UnivariateContinuousDistribution.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ProbabilityDensityFunction	Gets the probability density function (pdf) for this distribution evaluated at point x. (Inherited from UnivariateContinuousDistribution.)
	QuantileDensityFunction	Gets the first derivative of the inverse distribution function (icdf) for this distribution evaluated at probability p. (Inherited from UnivariateContinuousDistribution.)
	ToString	Returns a String that represents this instance. (Inherited from DistributionBase.)
	ToString(IFormatProvider)	Returns a String that represents this instance. (Inherited from DistributionBase.)
	ToString(String)	Returns a String that represents this instance. (Inherited from DistributionBase.)
	ToString(String, IFormatProvider)	Returns a String that represents this instance. (Overrides DistributionBaseToString(String, IFormatProvider).)

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Extension Methods

	Name	Description
	HasMethod	Checks whether an object implements a method with the given name. (Defined by ExtensionMethods.)
	IsEqual	Compares two objects for equality, performing an elementwise comparison if the elements are vectors or matrices. (Defined by Matrix.)
	To(Type)	Overloaded. Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime. (Defined by ExtensionMethods.)
	ToT	Overloaded. Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime. (Defined by ExtensionMethods.)

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Remarks

In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another distribution. It is specified by three parameters: location μ, scale σ, and shape ξ. Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as κ = − ξ.

References:

Wikipedia, The Free Encyclopedia. Generalized Pareto distribution. Available from: https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

Examples

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// Create a new Generalized Pareto Distribution with mu = 0, sigma = 0.42, xi = 3
var pareto = new GeneralizedParetoDistribution(location: 0, scale: 0.42, shape: 3);

// Common measures
double mean = pareto.Mean;     // -0.21
double median = pareto.Median; //  0.98
double var = pareto.Variance;  // -0.00882
double mode = pareto.Mode;     //  0.19185185523755152

// Cumulative distribution functions
double cdf = pareto.DistributionFunction(x: 1.4); // 0.55035568697739079
double ccdf = pareto.ComplementaryDistributionFunction(x: 1.4); // 0.44964431302260921
double icdf = pareto.InverseDistributionFunction(p: cdf); // 1.3999999035548829

// Probability density functions
double pdf = pareto.ProbabilityDensityFunction(x: 1.4); // 0.097325608879352654
double lpdf = pareto.LogProbabilityDensityFunction(x: 1.4); // -2.3296931293597707

// Hazard (failure rate) functions
double hf = pareto.HazardFunction(x: 1.4); // 0.21645021645021648
double chf = pareto.CumulativeHazardFunction(x: 1.4); // 0.79929842426612341

// String representation
string str = pareto.ToString(CultureInfo.InvariantCulture); // Pareto(x; μ = 0, σ = 0.42, ξ = 3)

Reference

Accord.Statistics.Distributions.Univariate Namespace

Accord.Statistics.Distributions.UnivariateParetoDistribution

Accord.Statistics.Distributions.UnivariateUnivariateContinuousDistribution