Create a Generalised Pareto (GP) distribution

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.

Examples


set.seed(27)

X <- GP(0, 2, 0.1)
X
#> [1] "GP(mu = 0, sigma = 2, xi = 0.1)"

random(X, 10)
#>  [1] 8.571201574 0.175715851 4.600737645 0.814822940 0.509138521 1.053986338
#>  [7] 0.151089620 0.004907082 0.297083889 0.430734122

pdf(X, 0.7)
#> [1] 0.3424729
log_pdf(X, 0.7)
#> [1] -1.071563

cdf(X, 0.7)
#> [1] 0.2910812
quantile(X, 0.7)
#> [1] 2.558897

cdf(X, quantile(X, 0.7))
#> [1] 0.7
quantile(X, cdf(X, 0.7))
#> [1] 0.7