# 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
kShape parameter$-\infty \le k\le \infty$
sigmaScale parameter$\sigma \ge 0$
muLocation parameter$-\infty \le \mu \le \infty$

## Creation

There are several ways to create a GeneralizedExtremeValueDistribution 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 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

This property is read-only.

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

This property is read-only.

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

Data Types: double

This property is read-only.

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

This property is read-only.

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

This property is read-only.

Distribution parameter values, specified as a vector.

Data Types: single | double

This property is read-only.

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

Data Types: single | double

### Other Object Properties

This property is read-only.

Probability distribution name, specified as a character vector.

Data Types: char

This property is read-only.

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

This property is read-only.

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

Data Types: char

This property is read-only.

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 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

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