The Gumbel distribution

The Gumbel distribution is a special case of the Generalized Extreme Value distribution, obtained when the GEV shape parameter \(\xi\) is equal to 0. It may be referred to as a type I extreme value distribution.

dist_gumbel(alpha, scale)

Arguments

alpha

location parameter.

scale

parameter. Must be strictly positive.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_gumbel.html

In the following, let \(X\) be a Gumbel random variable with location parameter alpha = \(\alpha\) and scale parameter scale = \(\sigma\).

Support: \(R\), the set of all real numbers.

Mean:

$$ E(X) = \alpha + \sigma\gamma $$

where \(\gamma\) is the Euler-Mascheroni constant, approximately equal to 0.5772157.

Variance:

$$ \textrm{Var}(X) = \frac{\pi^2 \sigma^2}{6} $$

Skewness:

$$ \textrm{Skew}(X) = \frac{12\sqrt{6}\zeta(3)}{\pi^3} \approx 1.1395 $$

where \(\zeta(3)\) is Apery's constant, approximately equal to 1.2020569. Note that skewness is independent of the distribution parameters.

Kurtosis (excess):

$$ \textrm{Kurt}(X) = \frac{12}{5} = 2.4 $$

Note that excess kurtosis is independent of the distribution parameters.

Median:

$$ \textrm{Median}(X) = \alpha - \sigma\ln(\ln 2) $$

Probability density function (p.d.f):

$$ f(x) = \frac{1}{\sigma} \exp\left[-\frac{x - \alpha}{\sigma}\right] \exp\left\{-\exp\left[-\frac{x - \alpha}{\sigma}\right]\right\} $$

for \(x\) in \(R\), the set of all real numbers.

Cumulative distribution function (c.d.f):

$$ F(x) = \exp\left\{-\exp\left[-\frac{x - \alpha}{\sigma}\right]\right\} $$

for \(x\) in \(R\), the set of all real numbers.

Quantile function (inverse c.d.f):

$$ F^{-1}(p) = \alpha - \sigma \ln(-\ln p) $$

for \(p\) in (0, 1).

Moment generating function (m.g.f):

$$ E(e^{tX}) = \Gamma(1 - \sigma t) e^{\alpha t} $$

for \(\sigma t < 1\), where \(\Gamma\) is the gamma function.

See also

actuar::Gumbel, actuar::dgumbel(), actuar::pgumbel(), actuar::qgumbel(), actuar::rgumbel(), actuar::mgumbel()

Examples

dist <- dist_gumbel(alpha = c(0.5, 1, 1.5, 3), scale = c(2, 2, 3, 4))
dist
#> <distribution[4]>
#> [1] Gumbel(0.5, 2) Gumbel(1, 2)   Gumbel(1.5, 3) Gumbel(3, 4)  

mean(dist)
#> [1] 1.654431 2.154431 3.231647 5.308863
variance(dist)
#> [1]  6.579736  6.579736 14.804407 26.318945
skewness(dist)
#> [1] 1.139547 1.139547 1.139547 1.139547
kurtosis(dist)
#> [1] 2.4 2.4 2.4 2.4
support(dist)
#> <support_region[4]>
#> [1] R R R R
generate(dist, 10)
#> [[1]]
#>  [1]  1.4353349  2.9955069 -1.0627131  2.0642517  2.6329347  1.3938184
#>  [7]  2.6516692  2.9493220  0.9966659  1.3503559
#> 
#> [[2]]
#>  [1]  3.86462213  1.76513625  0.65753682 -0.47749966  1.56383324  0.29691494
#>  [7]  2.89096392 -0.09381715  2.61300191  1.68342665
#> 
#> [[3]]
#>  [1]  8.48714227  0.49592489  2.69556134  1.86357146  0.90723133  0.04270603
#>  [7]  8.27850253  5.07597183  2.12817930 -0.04058970
#> 
#> [[4]]
#>  [1]  7.5231271  2.7249391 14.9064583  8.7440753  0.6090924  5.4986474
#>  [7]  4.8317315  0.3431633  1.2244111  1.8830472
#> 

density(dist, 2)
#> [1] 0.14726616 0.16535215 0.12102469 0.08889319
density(dist, 2, log = TRUE)
#> [1] -1.915514 -1.799678 -2.111761 -2.420320

cdf(dist, 4)
#> [1] 0.8404869 0.8000107 0.6475248 0.4589561

quantile(dist, 0.7)
#> [1] 2.561861 3.061861 4.592791 7.123722