# GeneralizedExtremeValueDistribution

Generalized extreme value probability distribution object

## Description

A `GeneralizedExtremeValueDistribution` object consists of parameters, a model description, and sample data for a generalized extreme value probability distribution.

The generalized extreme value distribution is often used to model the smallest or largest value among a large set of independent, identically distributed random values representing measurements or observations. It combines three simpler distributions into a single form, allowing a continuous range of possible shapes that include all three of the simpler distributions.

The three distribution types correspond to the limiting distribution of block maxima from different classes of underlying distributions:

• Type 1 — Distributions whose tails decrease exponentially, such as the normal distribution

• Type 2 — Distributions whose tails decrease as a polynomial, such as Student’s t distribution

• Type 3 — Distributions whose tails are finite, such as the beta distribution

The generalized extreme value distribution uses the following parameters.

ParameterDescriptionSupport
`k`Shape parameter$-\infty \le k\le \infty$
`sigma`Scale parameter$\sigma \ge 0$
`mu`Location parameter$-\infty \le \mu \le \infty$

## Creation

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

## Properties

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### Distribution Parameters

Shape parameter of the generalized extreme value distribution, specified as a scalar value.

Data Types: `single` | `double`

Scale parameter of the generalized extreme value distribution, specified as a nonnegative scalar value.

Data Types: `single` | `double`

Location parameter of the generalized extreme value 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 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 extreme value distribution object using the default parameter values.

`pd = makedist('GeneralizedExtremeValue')`
```pd = GeneralizedExtremeValueDistribution Generalized Extreme Value distribution k = 0 sigma = 1 mu = 0 ```

Create a generalized extreme value distribution object by specifying values for the parameters.

`pd = makedist('GeneralizedExtremeValue','k',0,'sigma',2,'mu',1)`
```pd = GeneralizedExtremeValueDistribution Generalized Extreme Value distribution k = 0 sigma = 2 mu = 1 ```

Compute the mean of the distribution.

`m = mean(pd)`
```m = 2.1544 ```

## Version History

Introduced in R2013a