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