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# prob.GeneralizedExtremeValueDistribution class

Package: prob
Superclasses: prob.ToolboxFittableParametricDistribution

Generalized extreme value probability distribution object

## Description

`prob.GeneralizedExtremeValueDistribution` is an object consisting of parameters, a model description, and sample data for a generalized extreme value 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('GeneralizedExtremeValue')` creates a generalized extreme value probability distribution object using the default parameter values.

`pd = makedist('GeneralizedExtremeValue','k',k,'sigma',sigma,'mu',mu)` creates a generalized extreme value probability distribution object using the specified parameter values.

### Input Arguments

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

Data Types: `single` | `double`

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

Data Types: `single` | `double`

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

Data Types: `single` | `double`

## Properties

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

Data Types: `single` | `double`

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

Data Types: `single` | `double`

Location parameter of the generalized extreme value distribution, stored as a scalar value.

Data Types: `single` | `double`

Probability distribution name, stored as a character vector. This property is read-only.

Data Types: `char`

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.

Data Types: `struct`

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`

Number of parameters for the probability distribution, stored as a positive integer value. This property is read-only.

Data Types: `single` | `double`

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 `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. This property is read-only.

Data Types: `single` | `double`

Distribution parameter descriptions, stored as a cell array of character vectors. Each cell contains a short description of one distribution parameter. This property is read-only.

Data Types: `char`

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`

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

Data Types: `char`

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

Data Types: `single` | `double`

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 log likelihood 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 Extreme Value 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$-\infty \le \mu \le \infty$

The probability density function (pdf) for a Type 1 distribution, where shape parameter $k=0$, is

`$f\left(x|0,\mu ,\sigma \right)=\left(\frac{1}{\sigma }\right)\mathrm{exp}\left(-\mathrm{exp}\left(-\frac{\left(x-\mu \right)}{\sigma }\right)-\frac{\left(x-\mu \right)}{\sigma }\right)\text{ };\text{ }-\infty `

When $k\ne 0$, the pdf is

`$f\left(x|k,\mu ,\sigma \right)=\left(\frac{1}{\sigma }\right)\mathrm{exp}\left(-{\left(1+k\frac{\left(x-\mu \right)}{\sigma }\right)}^{-\frac{1}{k}}\right){\left(1+k\frac{\left(x-\mu \right)}{\sigma }\right)}^{-1-\frac{1}{k}}$`

for

`$1+k\frac{\left(x-\mu \right)}{\sigma }>0.$`

For the Type 2 case, $k>0$ and $x\ge \mu -\frac{\sigma }{k}$. For the Type 3 case, $k<0$ and $x<\mu -\frac{\sigma }{k}$.

## 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```