The GP distribution has a link to the \link{GEV}
distribution.
Suppose that the maximum of \(n\) i.i.d. random variables has
approximately a GEV distribution. For a sufficiently large threshold
\(u\), the conditional distribution of the amount (the threshold
excess) by which a variable exceeds \(u\) given that it exceeds \(u\)
has approximately a GP distribution. Therefore, the GP distribution is
often used to model the threshold excesses of a high threshold \(u\).
The requirement that the variables are independent can be relaxed
substantially, but then exceedances of \(u\) may cluster.
GP(mu = 0, sigma = 1, xi = 0)
A GP
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 GP random variable with location
parameter mu
= \(\mu\), scale parameter sigma
= \(\sigma\) and
shape parameter xi
= \(\xi\).
Support: \([\mu, \mu - \sigma / \xi]\) for \(\xi < 0\); \([\mu, \infty)\) for \(\xi \geq 0\).
Mean: \(\mu + \sigma/(1 - \xi)\) for \(\xi < 1\); undefined otherwise.
Median: \(\mu + \sigma[2 ^ \xi - 1]/\xi\) for \(\xi \neq 0\); \(\mu + \sigma\ln 2\) for \(\xi = 0\).
Variance: \(\sigma^2 / (1 - \xi)^2 (1 - 2\xi)\) for \(\xi < 1 / 2\); undefined otherwise.
Probability density function (p.d.f):
If \(\xi \neq 0\) then $$f(x) = \sigma^{-1} [1 + \xi (x - \mu) / \sigma] ^ {-(1 + 1/\xi)}$$ for \(1 + \xi (x - \mu) / \sigma > 0\). The p.d.f. is 0 outside the support.
In the \(\xi = 0\) special case $$f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]$$ for \(x\) in [\(\mu, \infty\)). The p.d.f. is 0 outside the support.
Cumulative distribution function (c.d.f):
If \(\xi \neq 0\) then $$F(x) = 1 - \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}$$ for \(1 + \xi (x - \mu) / \sigma > 0\). The c.d.f. is 0 below the support and 1 above the support.
In the \(\xi = 0\) special case $$F(x) = 1 - \exp[-(x - \mu) / \sigma] \}$$ for \(x\) in \(R\), the set of all real numbers.
set.seed(27)
X <- GP(0, 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))
Run the code above in your browser using DataLab