Documentation

# 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
`k`Shape parameter$-\infty
`sigma`Scale parameter$\sigma \ge 0$
`theta`Location parameter$-\infty <\theta <\infty$

## Creation

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

## 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

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 `i`th parameter and the `j`th parameter. The (`i`,`i`) element is the estimated variance of the `i`th 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.

Data Types: `single` | `double`

Truncation interval for the probability distribution, specified as a vector 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 `icdf` Inverse cumulative distribution function `iqr` Interquartile range `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 `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 ```