# Frechet: Frechet Distribution Class In distr6: The Complete R6 Probability Distributions Interface

 Frechet R Documentation

## Frechet Distribution Class

### Description

Mathematical and statistical functions for the Frechet distribution, which is commonly used as a special case of the Generalised Extreme Value distribution.

### Details

The Frechet distribution parameterised with shape, α, scale, β, and minimum, γ, is defined by the pdf,

f(x) = (α/β)((x-γ)/β)^{-1-α}exp(-(x-γ)/β)^{-α}

for α, β ε R^+ and γ ε R.

### Value

Returns an R6 object inheriting from class SDistribution.

### Distribution support

The distribution is supported on x > γ.

### Default Parameterisation

Frec(shape = 1, scale = 1, minimum = 0)

N/A

### Also known as

Also known as the Inverse Weibull distribution.

### Super classes

distr6::Distribution -> distr6::SDistribution -> Frechet

### Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

packages

Packages required to be installed in order to construct the distribution.

### Active bindings

properties

Returns distribution properties, including skewness type and symmetry.

### Methods

#### Public methods

Inherited methods

#### Method new()

Creates a new instance of this R6 class.

##### Usage
Frechet\$new(shape = NULL, scale = NULL, minimum = NULL, decorators = NULL)
##### Arguments
shape

(numeric(1))
Shape parameter, defined on the positive Reals.

scale

(numeric(1))
Scale parameter, defined on the positive Reals.

minimum

(numeric(1))
Minimum of the distribution, defined on the Reals.

decorators

(character())
Decorators to add to the distribution during construction.

#### Method mean()

The arithmetic mean of a (discrete) probability distribution X is the expectation

E_X(X) = ∑ p_X(x)*x

with an integration analogue for continuous distributions.

##### Usage
Frechet\$mean(...)
...

Unused.

#### Method mode()

The mode of a probability distribution is the point at which the pdf is a local maximum, a distribution can be unimodal (one maximum) or multimodal (several maxima).

##### Usage
Frechet\$mode(which = "all")
##### Arguments
which

(character(1) | numeric(1)
Ignored if distribution is unimodal. Otherwise "all" returns all modes, otherwise specifies which mode to return.

#### Method median()

Returns the median of the distribution. If an analytical expression is available returns distribution median, otherwise if symmetric returns self\$mean, otherwise returns self\$quantile(0.5).

Frechet\$median()

#### Method variance()

The variance of a distribution is defined by the formula

var_X = E[X^2] - E[X]^2

where E_X is the expectation of distribution X. If the distribution is multivariate the covariance matrix is returned.

##### Usage
Frechet\$variance(...)
...

Unused.

#### Method skewness()

The skewness of a distribution is defined by the third standardised moment,

sk_X = E_X[((x - μ)/σ)^3]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution.

##### Usage
Frechet\$skewness(...)
...

Unused.

#### Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment,

k_X = E_X[((x - μ)/σ)^4]

where E_X is the expectation of distribution X, μ is the mean of the distribution and σ is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

##### Usage
Frechet\$kurtosis(excess = TRUE, ...)
##### Arguments
excess

(logical(1))
If TRUE (default) excess kurtosis returned.

...

Unused.

#### Method entropy()

The entropy of a (discrete) distribution is defined by

- ∑ (f_X)log(f_X)

where f_X is the pdf of distribution X, with an integration analogue for continuous distributions.

##### Usage
Frechet\$entropy(base = 2, ...)
##### Arguments
base

(integer(1))
Base of the entropy logarithm, default = 2 (Shannon entropy)

...

Unused.

#### Method pgf()

The probability generating function is defined by

pgf_X(z) = E_X[exp(z^x)]

where X is the distribution and E_X is the expectation of the distribution X.

##### Usage
Frechet\$pgf(z, ...)
##### Arguments
z

(integer(1))
z integer to evaluate probability generating function at.

...

Unused.

#### Method clone()

The objects of this class are cloneable with this method.

##### Usage
Frechet\$clone(deep = FALSE)
##### Arguments
deep

Whether to make a deep clone.

### References

McLaughlin, M. P. (2001). A compendium of common probability distributions (pp. 2014-01). Michael P. McLaughlin.