distributions3 (version 0.1.2)

GP: Create a Generalised Pareto (GP) distribution

Description

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.

Usage

GP(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 GP 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 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.

See Also

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

Examples

# NOT RUN {
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))
# }