View source: R/GeneralisedExtremeValue.R
GEV | R Documentation |
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.
GEV(mu = 0, sigma = 1, xi = 0)
mu |
The location parameter, written μ in textbooks.
|
sigma |
The scale parameter, written σ in textbooks.
|
xi |
The shape parameter, written ξ in textbooks.
|
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
= μ, scale parameter sigma
= σ and
shape parameter xi
= ξ.
Support: (-∞, μ - σ / ξ) for ξ < 0; (μ - σ / ξ, ∞) for ξ > 0; and R, the set of all real numbers, for ξ = 0.
Mean: μ + σ[Γ(1 - ξ) - 1]/ξ for ξ < 1, ξ != 0; μ + σγ for ξ = 0, where γ is Euler's constant, approximately equal to 0.57722; undefined otherwise.
Median: μ + σ[(ln 2)^(-ξ) - 1] / ξ for ξ != 0; μ - σ ln(ln 2) for ξ = 0.
Variance: σ^2 [Γ(1 - 2 ξ) - Γ(1 - ξ)^2] / ξ^2 for ξ < 1 / 2, ξ != 0; σ^2 π^2 / 6 for ξ = 0; undefined otherwise.
Probability density function (p.d.f):
If ξ is not equal to 0 then
f(x) = (1 / σ) [1 + ξ (x - μ) / σ] ^ {-(1 + 1/ξ)} exp{ -[1 + ξ (x - μ) / σ] ^ (-1/ξ)}
for 1 + ξ (x - μ) / σ > 0. The p.d.f. is 0 outside the support.
In the ξ = 0 (Gumbel) special case
f(x) = (1 / σ) exp[-(x - μ) / σ] exp{-exp[-(x - μ) / σ]}
for x in R, the set of all real numbers.
Cumulative distribution function (c.d.f):
If ξ is not equal to 0 then
F(x) = exp{ -[1 + ξ (x - μ) / σ] ^ (-1/ξ)}
for 1 + ξ (x - μ) / σ > 0. The c.d.f. is 0 below the support and 1 above the support.
In the ξ = 0 (Gumbel) special case
F(x) = exp{ - exp[-(x - μ) / σ]}
for x in R, the set of all real numbers.
A GEV
object.
Other continuous distributions:
Beta()
,
Cauchy()
,
ChiSquare()
,
Erlang()
,
Exponential()
,
FisherF()
,
Frechet()
,
GP()
,
Gamma()
,
Gumbel()
,
LogNormal()
,
Logistic()
,
Normal()
,
RevWeibull()
,
StudentsT()
,
Tukey()
,
Uniform()
,
Weibull()
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))
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