GeneralizedParetoDistribution

Generalized Pareto probability distribution object

Description

A GeneralizedParetoDistribution object consists of parameters, a model description, and sample data for a generalized Pareto probability 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 exceedance 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<

Creation

There are several ways to create a GeneralizedParetoDistribution probability distribution object.

  • Create a distribution with specified parameter values using makedist.

  • Fit a distribution to data using fitdist.

  • Interactively fit a distribution to data using the Distribution Fitter app.

Properties

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Distribution Parameters

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

Data Types: single | double

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

Data Types: single | double

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

Data Types: single | double

Distribution Characteristics

This property is read-only.

Logical flag for truncated distribution, specified as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated.

Data Types: logical

This property is read-only.

Number of parameters for the probability distribution, specified as a positive integer value.

Data Types: double

This property is read-only.

Covariance matrix of the parameter estimates, specified 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.

Data Types: double

This property is read-only.

Logical flag for fixed parameters, specified 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.

Data Types: logical

This property is read-only.

Distribution parameter values, specified as a vector.

Data Types: single | double

This property is read-only.

Truncation interval for the probability distribution, specified as a vector containing the lower and upper truncation boundaries.

Data Types: single | double

Other Object Properties

This property is read-only.

Probability distribution name, specified as a character vector.

Data Types: char

This property is read-only.

Data used for distribution fitting, specified 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.

Data Types: struct

This property is read-only.

Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: char

This property is read-only.

Distribution parameter names, specified as a cell array of character vectors.

Data Types: char

Object Functions

cdfCumulative distribution function
icdfInverse cumulative distribution function
iqrInterquartile range
meanMean of probability distribution
medianMedian of probability distribution
negloglikNegative loglikelihood of probability distribution
paramciConfidence intervals for probability distribution parameters
pdfProbability density function
proflikProfile likelihood function for probability distribution
randomRandom numbers
stdStandard deviation of probability distribution
truncateTruncate probability distribution object
varVariance of probability distribution

Examples

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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 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 = 3

Introduced in R2013a