GEV function

Generalized extreme value distribution

Generalized extreme value distribution

Density, distribution function, quantile function and random generation for the generalized extreme value distribution.

dgev(x, mu = 0, sigma = 1, xi = 0, log = FALSE)

pgev(q, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

qgev(p, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

rgev(n, mu = 0, sigma = 1, xi = 0)

Arguments

  • x, q: vector of quantiles.

  • mu, sigma, xi: location, scale, and shape parameters. Scale must be positive.

  • log, log.p: logical; if TRUE, probabilities p are given as log(p).

  • lower.tail: logical; if TRUE (default), probabilities are P[X≤x]P[X \le x]

    otherwise, P[X>x]P[X > x].

  • p: vector of probabilities.

  • n: number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x)={1σ(1+ξx−μσ)−1/ξ−1exp⁡(−(1+ξx−μσ)−1/ξ)ξ≠01σexp⁡(−x−μσ)exp⁡(−exp⁡(−x−μσ))ξ=0f(x)=[ifξ!=0:]1/σ∗(1+ξ∗(x−μ)/σ)−1/ξ−1∗exp(−(1+ξ∗(x−μ)/σ)−1/ξ)[else:]1/σ∗exp(−(x−μ)/σ)∗exp(−exp(−(x−μ)/σ)) f(x) = \left\{\begin{array}{ll}\frac{1}{\sigma} \left(1+\xi \frac{x-\mu}{\sigma}\right)^{-1/\xi-1} \exp\left(-\left(1+\xi \frac{x-\mu}{\sigma}\right)^{-1/\xi}\right) & \xi \neq 0 \\\frac{1}{\sigma} \exp\left(- \frac{x-\mu}{\sigma}\right) \exp\left(-\exp\left(- \frac{x-\mu}{\sigma}\right)\right) & \xi = 0\end{array}\right.f(x) = [if \xi != 0:] 1/\sigma * (1+\xi*(x-\mu)/\sigma)^{-1/\xi-1} * exp(-(1+\xi*(x-\mu)/\sigma)^{-1/\xi})[else:] 1/\sigma * exp(-(x-\mu)/\sigma) * exp(-exp(-(x-\mu)/\sigma))

Cumulative distribution function

F(x)={exp⁡(−(1+ξx−μσ)1/ξ)ξ≠0exp⁡(−exp⁡(−x−μσ))ξ=0F(x)=[ifξ!=0:]exp(−(1+ξ∗(x−μ)/σ)1/ξ)[else:]exp(−exp(−(x−μ)/σ)) F(x) = \left\{\begin{array}{ll}\exp\left(-\left(1+\xi \frac{x-\mu}{\sigma}\right)^{1/\xi}\right) & \xi \neq 0 \\\exp\left(-\exp\left(- \frac{x-\mu}{\sigma}\right)\right) & \xi = 0\end{array}\right.F(x) = [if \xi != 0:] exp(-(1+\xi*(x-\mu)/\sigma)^{1/\xi})[else:] exp(-exp(-(x-\mu)/\sigma))

Quantile function

F−1(p)={μ−σξ(1−(−log⁡(p))ξ)ξ≠0μ−σlog⁡(−log⁡(p))ξ=0F−1(p)=[ifξ!=0:]μ−σ/ξ∗(1−(−log(p))ξ)[else:]μ−σ∗log(−log(p)) F^{-1}(p) = \left\{\begin{array}{ll}\mu - \frac{\sigma}{\xi} (1 - (-\log(p))^\xi) & \xi \neq 0 \\\mu - \sigma \log(-\log(p)) & \xi = 0\end{array}\right.F^-1(p) = [if \xi != 0:] \mu - \sigma/\xi * (1 - (-log(p))^\xi)[else:] \mu - \sigma * log(-log(p))

Examples

curve(dgev(x, xi = -1/2), -4, 4, col = "green", ylab = "")
curve(dgev(x, xi = 0), -4, 4, col = "red", add = TRUE)
curve(dgev(x, xi = 1/2), -4, 4, col = "blue", add = TRUE)
legend("topleft", col = c("green", "red", "blue"), lty = 1,
       legend = expression(xi == -1/2, xi == 0, xi == 1/2), bty = "n")

x <- rgev(1e5, 5, 2, .5)
hist(x, 1000, freq = FALSE, xlim = c(0, 50))
curve(dgev(x, 5, 2, .5), 0, 50, col = "red", add = TRUE, n = 5000)
hist(pgev(x, 5, 2, .5))
plot(ecdf(x), xlim = c(0, 50))
curve(pgev(x, 5, 2, .5), 0, 50, col = "red", lwd = 2, add = TRUE)

References

Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.

extraDistr packageRead PDF manual
  • Maintainer: Tymoteusz Wolodzko
  • License: GPL-2
  • Last published: 2023-11-30

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