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# 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.

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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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### k — Shape parameterscalar value

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

Data Types: single | double

### sigma — Scale parameternonnegative scalar value

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

Data Types: single | double

### theta — Location parameterscalar value

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

Data Types: single | double

### DistributionName — Probability distribution nameprobability distribution name string

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

Data Types: char

### InputData — Data 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

### IsTruncated — Logical 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

### NumParameters — Number 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

### ParameterCovariance — Covariance 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

### ParameterDescription — Distribution 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

### ParameterIsFixed — Logical 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

### ParameterNames — Distribution parameter namescell array of strings

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

Data Types: char

### ParameterValues — Distribution parameter valuesvector of scalar values

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

Data Types: single | double

### Truncation — Truncation 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 icdf Inverse cumulative distribution function of probability distribution object iqr Interquartile range of probability distribution object median Median of probability distribution object pdf Probability density function of probability distribution object random Generate random numbers from probability distribution object truncate Truncate probability distribution object
 mean Mean of probability distribution object negloglik Negative loglikelihood of probability distribution object paramci Confidence intervals for probability distribution parameters proflik Profile likelihood function for probability distribution object std Standard deviation of probability distribution object var Variance 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$-\infty
sigmaScale parameter$\sigma \ge 0$
thetaLocation parameter$-\infty <\theta <\infty$

The probability density function (pdf) of the generalized Pareto distribution with shape parameter $k\ne 0$ is

$f\left(x|k,\sigma ,\theta \right)=\left(\frac{1}{\sigma }\right)\text{\hspace{0.17em}}{\left(1+k\frac{\left(x-\theta \right)}{\sigma }\right)}^{-1-\frac{1}{k}}$

for $x>\theta$, when $k>0$, or for $\theta , when $k<0$.

For $k=0$, the pdf is

$y=f\left(x|0,\sigma ,\theta \right)=\left(\frac{1}{\sigma }\right)\mathrm{exp}\left(-\frac{\left(x-\theta \right)}{\sigma }\right)$

for $x>\theta$.

If $k=0$ and $\theta =0$, the generalized Pareto distribution is equivalent to the exponential distribution. If $k>0$ and $\theta =\frac{\sigma }{k}$, 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```