# GEV: Generalized extreme value distribution In extraDistr: Additional Univariate and Multivariate Distributions

## Description

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

## Usage

 ```1 2 3 4 5 6 7``` ```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] otherwise, 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) = [if ξ != 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))

Cumulative distribution function

F(x) = [if ξ != 0:] exp(-(1+ξ*(x-μ)/σ)^{1/ξ}) [else:] exp(-exp(-(x-μ)/σ))

Quantile function

F^-1(p) = [if ξ != 0:] μ - σ/ξ * (1 - (-log(p))^ξ) [else:] μ - σ * log(-log(p))

## References

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12``` ```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) ```

extraDistr documentation built on Sept. 7, 2020, 5:09 p.m.