distributional (version 0.5.0)

dist_gumbel: The Gumbel distribution

Description

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.

Usage

dist_gumbel(alpha, scale)

Arguments

alpha

location parameter.

scale

parameter. Must be strictly positive.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

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

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

Mean: \(\mu + \sigma\gamma\), where \(\gamma\) is Euler's constant, approximately equal to 0.57722.

Median: \(\mu - \sigma\ln(\ln 2)\).

Variance: \(\sigma^2 \pi^2 / 6\).

Probability density function (p.d.f):

$$f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]% \exp\{-\exp[-(x - \mu) / \sigma] \}$$ for \(x\) in \(R\), the set of all real numbers.

Cumulative distribution function (c.d.f):

In the \(\xi = 0\) (Gumbel) special case $$F(x) = \exp\{-\exp[-(x - \mu) / \sigma] \}$$ for \(x\) in \(R\), the set of all real numbers.

See Also

actuar::Gumbel

Examples

dist <- dist_gumbel(alpha = c(0.5, 1, 1.5, 3), scale = c(2, 2, 3, 4))
dist

if (FALSE) { # requireNamespace("actuar", quietly = TRUE)
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)
}