# Gumbel distribution

### 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 Matthias.Kohl@stamats.de

### References

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

### See Also

`rgumbel`

, `AbscontDistribution-class`

### Examples

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