Gumbel Distribution

Two-parameter Gumbel (Extreme Value type I) distribution with location \(\mu\) and scale \(\sigma\). The Gumbel distribution is the limiting distribution of block maxima and is widely used in extreme-value analysis of hydroclimatic variables. Density, distribution, quantile, and random generation functions are provided via the FAdist package. Fitting is performed via closed-form L-moments through pelgum.

Define \(z = (x - \mu)/\sigma\). The probability density function is: $$f(x) = \frac{1}{\sigma} \exp(-z - e^{-z})$$ where:

• \(\mu\) — location parameter (location)

• \(\sigma\) — scale parameter (scale)

Usage

dgumbel(x, location, scale)

pgumbel(q, location, scale)

qgumbel(p, location, scale)

rgumbel(n, location, scale)

Arguments

x, q

vector of quantiles.

location

location parameter.

scale

scale parameter.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken as the number required.