Contents

prob.GeneralizedParetoDistribution class

Package: prob
Superclasses: prob.ToolboxFittableParametricDistribution

Generalized Pareto probability distribution object

Description

prob.GeneralizedParetoDistribution is an object consisting of parameters, a model description, and sample data for a generalized Pareto probability distribution.

Create a probability distribution object with specified parameter values using makedist. Alternatively, fit a distribution to data using fitdist or the Distribution Fitting app.

Construction

pd = makedist('GeneralizedParetoDistribution') creates a generalized Pareto probability distribution object using default parameter values.

pd = makedist('GeneralizedParetoDistribution','k',k,'sigma',sigma,'theta',theta) creates a generalized Pareto probability distribution object using the specified parameter values.

Input Arguments

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k — Shape parameter1 (default) | scalar value

Shape parameter for the generalized Pareto distribution, specified as a scalar value.

Data Types: single | double

sigma — Scale parameter1 (default) | nonnegative scalar value

Scale parameter for the generalized Pareto distribution, specified as a nonnegative scalar value.

Data Types: single | double

theta — Location parameter1 (default) | scalar value

Location parameter for the generalized Pareto distribution, specified as a scalar value.

Data Types: single | double

Properties

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kShape parameterscalar value

Shape parameter for the generalized Pareto distribution, stored as a scalar value.

Data Types: single | double

sigmaScale parameternonnegative scalar value

Scale parameter for the generalized Pareto distribution, stored as a nonnegative scalar value.

Data Types: single | double

thetaLocation parameterscalar value

Location parameter for the generalized Pareto distribution, stored as a scalar value.

Data Types: single | double

DistributionNameProbability distribution nameprobability distribution name string

Probability distribution name, stored as a valid probability distribution name string. This property is read-only.

Data Types: char

InputDataData used for distribution fittingstructure

Data used for distribution fitting, stored as a structure containing the following:

  • data: Data vector used for distribution fitting.

  • cens: Censoring vector, or empty if none.

  • freq: Frequency vector, or empty if none.

This property is read-only.

Data Types: struct

IsTruncatedLogical flag for truncated distribution0 | 1

Logical flag for truncated distribution, stored as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated. This property is read-only.

Data Types: logical

NumParametersNumber of parameterspositive integer value

Number of parameters for the probability distribution, stored as a positive integer value. This property is read-only.

Data Types: single | double

ParameterCovarianceCovariance matrix of the parameter estimatesmatrix of scalar values

Covariance matrix of the parameter estimates, stored as a p-by-p matrix, where p is the number of parameters in the distribution. The (i,j) element is the covariance between the estimates of the ith parameter and the jth parameter. The (i,i) element is the estimated variance of the ith parameter. If parameter i is fixed rather than estimated by fitting the distribution to data, then the (i,i) elements of the covariance matrix are 0. This property is read-only.

Data Types: single | double

ParameterDescriptionDistribution parameter descriptionscell array of strings

Distribution parameter descriptions, stored as a cell array of strings. Each cell contains a short description of one distribution parameter. This property is read-only.

Data Types: char

ParameterIsFixedLogical flag for fixed parametersarray of logical values

Logical flag for fixed parameters, stored as an array of logical values. If 0, the corresponding parameter in the ParameterNames array is not fixed. If 1, the corresponding parameter in the ParameterNames array is fixed. This property is read-only.

Data Types: logical

ParameterNamesDistribution parameter namescell array of strings

Distribution parameter names, stored as a cell array of strings. This property is read-only.

Data Types: char

ParameterValuesDistribution parameter valuesvector of scalar values

Distribution parameter values, stored as a vector. This property is read-only.

Data Types: single | double

TruncationTruncation intervalvector of scalar values

Truncation interval for the probability distribution, stored as a vector containing the lower and upper truncation boundaries. This property is read-only.

Data Types: single | double

Methods

Inherited Methods

cdf Cumulative distribution function of probability distribution object
icdfInverse cumulative distribution function of probability distribution object
iqrInterquartile range of probability distribution object
median Median of probability distribution object
pdfProbability density function of probability distribution object
randomGenerate random numbers from probability distribution object
truncateTruncate probability distribution object
meanMean of probability distribution object
negloglikNegative loglikelihood of probability distribution object
paramciConfidence intervals for probability distribution parameters
proflikProfile likelihood function for probability distribution object
std Standard deviation of probability distribution object
varVariance of probability distribution object

Definitions

Generalized Pareto Distribution

The generalized Pareto distribution is used to model the tails of another distribution. It allows a continuous range of possible shapes that include both the exponential and Pareto distributions as special cases. It has three basic forms, each corresponding to a limiting distribution of exceedence data from a different class of underlying distributions.

  • Distributions whose tails decrease exponentially, such as the normal, lead to a generalized Pareto shape parameter of zero.

  • Distributions whose tails decrease polynomially, such as the Student's t, lead to a positive shape parameter.

  • Distributions whose tails are finite, such as the beta, lead to a negative shape parameter.

The generalized Pareto distribution uses the following parameters.

ParameterDescriptionSupport
kShape parameter<k<
sigmaScale parameterσ0
thetaLocation parameter<θ<

The probability density function (pdf) of the generalized Pareto distribution with shape parameter k0 is

f(x|k,σ,θ)=(1σ)(1+k(xθ)σ)11k

for x>θ, when k>0, or for θ<x<σk, when k<0.

For k=0, the pdf is

y=f(x|0,σ,θ)=(1σ)exp((xθ)σ)

for x>θ.

If k=0 and θ=0, the generalized Pareto distribution is equivalent to the exponential distribution. If k>0 and θ=σk, the generalized Pareto distribution is equivalent to the Pareto distribution.

Examples

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Create a Generalized Pareto Distribution Object Using Default Parameters

Create a generalized Pareto distribution object using the default parameter values.

pd = makedist('GeneralizedPareto')
pd = 

  GeneralizedParetoDistribution

  Generalized Pareto distribution
        k = 1
    sigma = 1
    theta = 1

Create a Generalized Pareto Distribution Object Using Specified Parameters

Create a generalized Pareto distribution object by specifying parameter values.

pd = makedist('GeneralizedPareto','k',0,'sigma',2,'theta',1)
pd = 

  GeneralizedParetoDistribution

  Generalized Pareto distribution
        k = 0
    sigma = 2
    theta = 1

Compute the mean of the distribution.

m = mean(pd)
m =

    2.1544
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