distr6 (version 1.3.6)

Gumbel: Gumbel Distribution Class

Description

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.

Constructor

Gumbel$new(location = 0, scale = 1, decorators = NULL, verbose = FALSE)

Constructor Arguments

Argument	Type	Details
location	numeric	location parameter.
scale	numeric	scale parameter.

decorators Decorator decorators to add functionality. See details.

Constructor Details

The Gumbel distribution is parameterised with location as a numeric and scale as a positive numeric.

Public Variables

Variable	Return
name	Name of distribution.
short_name	Id of distribution.
description	Brief description of distribution.

Public Methods

Accessor Methods	Link
decorators	decorators
traits	traits
valueSupport	valueSupport
variateForm	variateForm
type	type
properties	properties
support	support
symmetry	symmetry
sup	sup
inf	inf
dmax	dmax
dmin	dmin
skewnessType	skewnessType
kurtosisType	kurtosisType

Statistical Methods Link pdf(x1, ..., log = FALSE, simplify = TRUE) pdf cdf(x1, ..., lower.tail = TRUE, log.p = FALSE, simplify = TRUE) cdf quantile(p, ..., lower.tail = TRUE, log.p = FALSE, simplify = TRUE) quantile.Distribution rand(n, simplify = TRUE) rand mean() mean.Distribution variance() variance stdev() stdev prec() prec cor() cor skewness() skewness kurtosis(excess = TRUE) kurtosis entropy(base = 2) entropy mgf(t) mgf cf(t) cf pgf(z) pgf median() median.Distribution iqr() iqr mode(which = "all") mode

Parameter Methods Link parameters(id) parameters getParameterValue(id, error = "warn") getParameterValue setParameterValue(..., lst = NULL, error = "warn") setParameterValue

Validation Methods Link liesInSupport(x, all = TRUE, bound = FALSE) liesInSupport liesInType(x, all = TRUE, bound = FALSE) liesInType

Representation Methods Link strprint(n = 2) strprint print(n = 2) print summary(full = T) summary.Distribution

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 \(\beta > 0\).

The distribution is supported on the Reals.

Apery's Constant to 16 significant figures is used in the skewness calculation. The gammaz function from the pracma package is used in the cf to allow complex inputs.

References

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

See Also

listDistributions for all available distributions. Frechet and Weibull for other special cases of the generalized extreme value distribution. gammaz for the references for the gamma function with complex inputs.

Examples

# NOT RUN {
x = Gumbel$new(location = 2, scale = 5)

# Update parameters
x$setParameterValue(scale = 3)
x$parameters()

# d/p/q/r
x$pdf(5)
x$cdf(5)
x$quantile(0.42)
x$rand(4)

# Statistics
x$mean()
x$variance()

summary(x)

# }