# Gumbel-class: Gumbel distribution In RobExtremes: Optimally Robust Estimation for Extreme Value Distributions

## Description

The Gumbel cumulative distribution function with location parameter `loc` = mu and scale parameter `scale` = sigma is

F(x) = exp(-exp[-(x-mu)/sigma])

for all real x, where sigma > 0; c.f. `rgumbel`. This distribution is also known as extreme value distribution of type I; confer Chapter~22 of Johnson et al. (1995).

## Objects from the Class

Objects can be created by calls of the form `new("Gumbel", loc, scale)`. More frequently they are created via the generating function `Gumbel`.

## Slots

`img`

Object of class `"Reals"`.

`param`

Object of class `"GumbelParameter"`.

`r`

`rgumbel`

`d`

`dgumbel`

`p`

`pgumbel`

`q`

`qgumbel`

`gaps`

(numeric) matrix or `NULL`

`.withArith`

logical: used internally to issue warnings as to interpretation of arithmetics

`.withSim`

logical: used internally to issue warnings as to accuracy

`.logExact`

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

`.lowerExact`

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

`Symmetry`

object of class `"DistributionSymmetry"`; used internally to avoid unnecessary calculations.

## Extends

Class `"AbscontDistribution"`, directly.
Class `"UnivariateDistribution"`, by class `"AbscontDistribution"`.
Class `"Distribution"`, by class `"AbscontDistribution"`.

## Methods

initialize

`signature(.Object = "Gumbel")`: initialize method.

loc

`signature(object = "Gumbel")`: wrapped access method for slot `loc` of slot `param`.

scale

`signature(x = "Gumbel")`: wrapped access method for slot `scale` of slot `param`.

loc<-

`signature(object = "Gumbel")`: wrapped replace method for slot `loc` of slot `param`.

scale<-

`signature(x = "Gumbel")`: wrapped replace method for slot `scale` of slot `param`.

`+`

`signature(e1 = "Gumbel", e2 = "numeric")`: result again of class `"Gumbel"`; exact.

`*`

`signature(e1 = "Gumbel", e2 = "numeric")`: result again of class `"Gumbel"`; exact.

E

`signature(object = "Gumbel", fun = "missing", cond = "missing")`: exact evaluation of expectation using explicit expressions.

var

`signature(x = "Gumbel")`: exact evaluation of expectation using explicit expressions.

skewness

`signature(x = "Gumbel")`: exact evaluation of expectation using explicit expressions.

kurtosis

`signature(x = "Gumbel")`: exact evaluation of expectation using explicit expressions.

median

`signature(x = "Gumbel")`: exact evaluation of expectation using explicit expressions.

IQR

`signature(x = "Gumbel")`: exact evaluation of expectation using explicit expressions.

## Note

This class is based on the code provided by the package evd.

## Author(s)

Matthias Kohl [email protected]

## References

Johnson et al. (1995) Continuous Univariate Distributions. Vol. 2. 2nd ed. New York: Wiley.

`rgumbel`, `AbscontDistribution-class`
 ```1 2 3 4 5 6 7``` ```(G1 <- new("Gumbel", loc = 1, scale = 2)) plot(G1) loc(G1) scale(G1) loc(G1) <- -1 scale(G1) <- 2 plot(G1) ```