Create a Generalised Extreme Value (GEV) distribution

The GEV distribution arises from the Extremal Types Theorem, which is rather like the Central Limit Theorem (see Normal()) but it relates to the maximum of $n$ i.i.d. random variables rather than to the sum. If, after a suitable linear rescaling, the distribution of this maximum tends to a non-degenerate limit as $n$ tends to infinity then this limit must be a GEV distribution. The requirement that the variables are independent can be relaxed substantially. Therefore, the GEV distribution is often used to model the maximum of a large number of random variables.

Usage

GEV(mu = 0, sigma = 1, xi = 0)

Arguments

mu: The location parameter, written $\mu$ in textbooks. mu can be any real number. Defaults to 0.
sigma: The scale parameter, written $\sigma$ in textbooks. sigma can be any positive number. Defaults to 1.
xi: The shape parameter, written $\xi$ in textbooks. xi can be any real number. Defaults to 0, which corresponds to a Gumbel distribution.

Value

A GEV object.

Details

We recommend reading this documentation on https://zeileis.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a GEV random variable with location parameter mu = $\mu$, scale parameter sigma = $\sigma$ and shape parameter xi = $\xi$.

Support: $(-\infty, \mu - \sigma / \xi)$ for $\xi < 0$; $(\mu - \sigma / \xi, \infty)$ for $\xi > 0$; and $R$, the set of all real numbers, for $\xi = 0$.

Mean: $\mu + \sigma[\Gamma(1 - \xi) - 1]/\xi$ for $\xi < 1, \xi \neq 0$; $\mu + \sigma\gamma$ for $\xi = 0$, where $\gamma$ is Euler's constant, approximately equal to 0.57722; undefined otherwise.

Median: $\mu + \sigma[(\ln 2) ^ {-\xi} - 1]/\xi$ for $\xi \neq 0$; $\mu - \sigma\ln(\ln 2)$ for $\xi = 0$.

Variance: $\sigma^2 [\Gamma(1 - 2 \xi) - \Gamma(1 - \xi)^2] / \xi^2$ for $\xi < 1 / 2, \xi \neq 0$; $\sigma^2 \pi^2 / 6$ for $\xi = 0$; undefined otherwise.

Probability density function (p.d.f):

If $\xi \neq 0$ then $$f(x) = \sigma ^ {-1} [1 + \xi (x - \mu) / \sigma] ^ {-(1 + 1/\xi)}% \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}$$ for $1 + \xi (x - \mu) / \sigma > 0$. The p.d.f. is 0 outside the support.

In the $\xi = 0$ (Gumbel) special case $$f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]% \exp\{-\exp[-(x - \mu) / \sigma] \}$$ for $x$ in $R$, the set of all real numbers.

Cumulative distribution function (c.d.f):

If $\xi \neq 0$ then $$F(x) = \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}$$ for $1 + \xi (x - \mu) / \sigma > 0$. The c.d.f. is 0 below the support and 1 above the support.

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

Examples


set.seed(27)

X <- GEV(1, 2, 0.1)
X
#> [1] "GEV(mu = 1, sigma = 2, xi = 0.1)"

random(X, 10)
#>  [1]  9.53039102 -0.73633998  5.43730770  0.79059280  0.20038342  1.18468635
#>  [7] -0.83938790 -2.28404509 -0.32725032  0.02226797

pdf(X, 0.7)
#> [1] 0.1845098
log_pdf(X, 0.7)
#> [1] -1.690052

cdf(X, 0.7)
#> [1] 0.3124986
quantile(X, 0.7)
#> [1] 3.171891

cdf(X, quantile(X, 0.7))
#> [1] 0.7
quantile(X, cdf(X, 0.7))
#> [1] 0.7