Gumbel Distribution

The Gumbel distribution, also known as the Type I extreme value distribution, is a continuous distribution used to model the maximum (or minimum) of a number of samples of various distributions. It is parameterized by location and scale and arises naturally in extreme value theory.

Usage

dgumbel(x, loc = 0, scale = 1, log = FALSE)

pgumbel(q, loc = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)

qgumbel(p, loc = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)

rgumbel(n, loc = 0, scale = 1)

Arguments

x, q: vector of quantiles.
loc, scale: location and scale parameters (can be given as vectors).
log, log.p: logical; if TRUE, probabilities p are given as log(p) and the density is returned on the log scale.
lower.tail: logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
p: vector of probabilities.
n: number of observations.

Value

dgumbel() gives the density function, pgumbel() gives the distribution function, qgumbel() gives the quantile function, and rgumbel() generates random deviates.

Details

Density function, distribution function, quantile function and random generation for the Gumbel distribution with location and scale parameters.

The Gumbel distribution function with parameters $`loc` = a$ and $`scale` = b$ is $$G(z) = \exp\left\{-\exp\left[-\left(\frac{z-a}{b}\right)\right]\right\}$$ for all real $z$, where $b > 0$.

Note

Based on code by Alec Stephenson previously published in the evd package, adapted to conform to package standards.

Examples


dgumbel(-1:2, -1, 0.5)
#> [1] 0.735758882 0.236409903 0.035966459 0.004945231
pgumbel(-1:2, -1, 0.5)
#> [1] 0.3678794 0.8734230 0.9818511 0.9975243
qgumbel(seq(0.9, 0.6, -0.1), 2, 0.5)
#> [1] 3.125184 2.749970 2.515465 2.335863
rgumbel(6, -1, 0.5)
#> [1] -1.093299 -1.017609 -1.609304 -1.179280 -1.258742 -1.460279
p <- (1:9)/10
pgumbel(qgumbel(p, -1, 2), -1, 2)
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9