distributions3 (version 0.2.2)

GEV: Create a Generalised Extreme Value (GEV) distribution

Description

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)

Value

A GEV object.

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.

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

Mean: $\mu +\sigma [\mathrm{\Gamma }(1-\xi )-1]/\xi$ for $\xi <1,\xi \ne 0$; $\mu +\sigma \gamma$ 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 $\xi \ne 0$; $\mu -\sigma \ln (\ln 2)$ for $\xi =0$.

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

Probability density function (p.d.f):

If $\xi \ne 0$ 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 $\xi \ne 0$ 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.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GP(), Gamma(), Gumbel(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))