GP: Create a Generalised Pareto (GP) distribution

In distributions3: Probability Distributions as S3 Objects

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 μ in textbooks. mu can be any real number. Defaults to 0.
sigma	The scale parameter, written σ in textbooks. sigma can be any positive number. Defaults to 1.
xi	The shape parameter, written ξ 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 GP random variable with location parameter mu = μ, scale parameter sigma = σ and shape parameter xi = ξ.

Support: [μ, μ - σ / ξ] for ξ < 0; [μ, ∞) for ξ >= 0.

Mean: μ + σ/(1 - ξ) for ξ < 1; undefined otherwise.

Median: μ + σ[2^ξ - 1] / ξ for ξ != 0; μ + σ ln2 for ξ = 0.

Variance: σ^2 / (1 - ξ)^2 (1 - 2ξ) for ξ < 1 / 2; 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)}

for 1 + ξ (x - μ) / σ > 0. The p.d.f. is 0 outside the support.

In the ξ = 0 special case

f(x) = (1 / σ) exp[-(x - μ) / σ]

for x in [μ, ∞). The p.d.f. is 0 outside the support.

Cumulative distribution function (c.d.f):

If ξ is not equal to 0 then

F(x) = 1 - 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 special case

F(x) = 1 - exp[-(x - μ) / σ]

for x in R, the set of all real numbers.

Value

A GP object.

See Also

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

Examples

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))

distributions3 documentation built on Sept. 7, 2022, 5:07 p.m.