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 \link{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://alexpghayes.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