# 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$-\infty
sigmaScale parameter$\sigma \ge 0$
thetaLocation (threshold) parameter$-\infty <\theta <\infty$

## 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 (threshold) parameter for the generalized Pareto distribution, specified as a scalar value.

Data Types: single | double

### Distribution Characteristics

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

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

Data Types: double

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

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

Distribution parameter values, specified as a vector of scalar values.

Data Types: single | double

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

Data Types: single | double

### Other Object Properties

Probability distribution name, specified as a character vector.

Data Types: char

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

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

Data Types: char

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

Data Types: char

## Object Functions

 cdf Cumulative distribution function gather Gather properties of Statistics and Machine Learning Toolbox object from GPU icdf Inverse cumulative distribution function iqr Interquartile range of probability distribution mean Mean of probability distribution median Median of probability distribution negloglik Negative loglikelihood of probability distribution paramci Confidence intervals for probability distribution parameters pdf Probability density function plot Plot probability distribution object proflik Profile likelihood function for probability distribution random Random numbers std Standard deviation of probability distribution truncate Truncate probability distribution object var Variance 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

## Version History

Introduced in R2013a