# gumbel

##### The Gumbel Distribution

Density, distribution function, quantile function, random generation, and gradient of density of the extreme value (maximum and minimum) distributions. The Gumbel distribution is also known as the extreme value maximum distribution, the double-exponential distribution and the log-Weibull distribution.

- Keywords
- distribution

##### Usage

`dgumbel(x, location = 0, scale = 1, log = FALSE, max = TRUE)`pgumbel(q, location = 0, scale = 1, lower.tail = TRUE, max = TRUE)

qgumbel(p, location = 0, scale = 1, lower.tail = TRUE, max = TRUE)

rgumbel(n, location = 0, scale = 1, max = TRUE)

ggumbel(x, max = TRUE)

##### Arguments

- x,q
numeric vector of quantiles.

- p
vector of probabilities.

- n
number of observations.

- location
numeric scalar.

- scale
numeric scalar.

- lower.tail
logical; if

`TRUE`

(default), probabilities are \(P[X \leq x]\) otherwise, \(P[X > x]\).- log
logical; if

`TRUE`

, probabilities p are given as log(p).- max
distribution for extreme maxima (default) or minima? The default corresponds to the standard right-skew Gumbel distribution.

##### Details

`dgumbel`

, `pgumbel`

and `ggumbel`

are implemented in C
for speed and care is taken that 'correct' results are provided for
values of `NA`

, `NaN`

, `Inf`

, `-Inf`

or just
extremely small or large.

The distribution functions, densities and gradients are used in the
Newton-Raphson algorithms in fitting cumulative link models with
`clm`

and cumulative link mixed models with
`clmm`

.

##### Value

`pgumbel`

gives the distribution function, `dgumbel`

gives the density, `ggumbel`

gives the gradient of the
density, `qgumbel`

is the quantile function, and
`rgumbel`

generates random deviates.

##### References

##### See Also

Gradients of densities are also implemented for the normal, logistic,
cauchy, cf. `gfun`

and the log-gamma distribution,
cf. `lgamma`

.

##### Examples

```
# NOT RUN {
## Illustrating the symmetry of the distribution functions:
pgumbel(5) == 1 - pgumbel(-5, max=FALSE) ## TRUE
dgumbel(5) == dgumbel(-5, max=FALSE) ## TRUE
ggumbel(5) == -ggumbel(-5, max=FALSE) ## TRUE
## More examples:
x <- -5:5
(pp <- pgumbel(x))
qgumbel(pp)
dgumbel(x)
ggumbel(x)
(ppp <- pgumbel(x, max=FALSE))
## Observe that probabilities close to 0 are more accurately determined than
## probabilities close to 1:
qgumbel(ppp, max=FALSE)
dgumbel(x, max=FALSE)
ggumbel(x, max=FALSE)
## random deviates:
set.seed(1)
(r1 <- rgumbel(10))
set.seed(1)
r2 <- -rgumbel(10, max = FALSE)
all(r1 == r2) ## TRUE
# }
```

*Documentation reproduced from package ordinal, version 2019.12-10, License: GPL (>= 2)*