Mathematical and statistical functions for the Gumbel distribution, which is commonly used to model the maximum (or minimum) of a number of samples of different distributions, and is a special case of the Generalised Extreme Value distribution.

Value

Returns an R6 object inheriting from class SDistribution.

Details

The Gumbel distribution parameterised with location, \(\mu\), and scale, \(\beta\), is defined by the pdf, $$f(x) = exp(-(z + exp(-z)))/\beta$$ for \(z = (x-\mu)/\beta\), \(\mu \epsilon R\) and \(\beta > 0\).

Distribution support

The distribution is supported on the Reals.

Default Parameterisation

Gumb(location = 0, scale = 1)

Omitted Methods

N/A

Also known as

N/A

References

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

Super classes

distr6::Distribution -> distr6::SDistribution -> Gumbel

Public fields

name

Full name of distribution.

short_name

Short name of distribution for printing.

description

Brief description of the distribution.

alias

Alias of the distribution.

packages

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

Methods

Inherited methods


Method new()

Creates a new instance of this R6 class.

Usage

Gumbel$new(location = NULL, scale = NULL, decorators = NULL)

Arguments

location

(numeric(1))
Location parameter defined on the Reals.

scale

(numeric(1))
Scale parameter defined on the positive 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 $$E_X(X) = \sum p_X(x)*x$$ with an integration analogue for continuous distributions.

Usage

Gumbel$mean(...)

Arguments

...

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

Gumbel$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).

Usage

Gumbel$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

Gumbel$variance(...)

Arguments

...

Unused.


Method skewness()

The skewness of a distribution is defined by the third standardised moment, $$sk_X = E_X[\frac{x - \mu}{\sigma}^3]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution.

Apery's Constant to 16 significant figures is used in the calculation.

Usage

Gumbel$skewness(...)

Arguments

...

Unused.


Method kurtosis()

The kurtosis of a distribution is defined by the fourth standardised moment, $$k_X = E_X[\frac{x - \mu}{\sigma}^4]$$ where \(E_X\) is the expectation of distribution X, \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation of the distribution. Excess Kurtosis is Kurtosis - 3.

Usage

Gumbel$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 $$- \sum (f_X)log(f_X)$$ where \(f_X\) is the pdf of distribution X, with an integration analogue for continuous distributions.

Usage

Gumbel$entropy(base = 2, ...)

Arguments

base

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

...

Unused.


Method mgf()

The moment generating function is defined by $$mgf_X(t) = E_X[exp(xt)]$$ where X is the distribution and \(E_X\) is the expectation of the distribution X.

Usage

Gumbel$mgf(t, ...)

Arguments

t

(integer(1))
t integer to evaluate function at.

...

Unused.


Method cf()

The characteristic function is defined by $$cf_X(t) = E_X[exp(xti)]$$ where X is the distribution and \(E_X\) is the expectation of the distribution X.

pracma::gammaz() is used in this function to allow complex inputs.

Usage

Gumbel$cf(t, ...)

Arguments

t

(integer(1))
t integer to evaluate function at.

...

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

Gumbel$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

Gumbel$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.