SpatialExtremes (version 2.1-0)

Generalized Pareto Distribution: The Generalized Pareto Distribution

Description

Density, distribution function, quantile function and random generation for the GP distribution with location equal to 'loc', scale equal to 'scale' and shape equal to 'shape'.

Usage

rgpd(n, loc = 0, scale = 1, shape = 0)
pgpd(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, lambda = 0)
qgpd(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, lambda = 0)
dgpd(x, loc = 0, scale = 1, shape = 0, log = FALSE)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations.

loc

vector of the location parameters.

scale

vector of the scale parameters.

shape

a numeric of the shape parameter.

lower.tail

logical; if TRUE (default), probabilities are \(\Pr[ X \le x]\), otherwise, \(\Pr[X > x]\).

log

logical; if TRUE, probabilities p are given as log(p).

lambda

a single probability - see the "value" section.

Value

If 'loc', 'scale' and 'shape' are not specified they assume the default values of '0', '1' and '0', respectively.

The GP distribution function for loc = \(u\), scale = \(\sigma\) and shape = \(\xi\) is

$$G(x) = 1 - \left[ 1 + \frac{\xi (x - u )}{ \sigma } \right] ^ { - 1 / \xi}$$

for \(1 + \xi ( x - u ) / \sigma > 0\) and \(x > u\), where \(\sigma > 0\). If \(\xi = 0\), the distribution is defined by continuity corresponding to the exponential distribution.

By definition, the GP distribution models exceedances above a threshold. In particular, the \(G\) function is a suited candidate to model

$$\Pr\left[ X \geq x | X > u \right] = 1 - G(x)$$ for \(u\) large enough.

However, it may be usefull to model the "non conditional" quantiles, that is the ones related to \(\Pr[ X \leq x]\). Using the conditional probability definition, one have :

$$\Pr\left[ X \geq x \right] = \left(1 - \lambda\right) \left( 1 + \xi \frac{x - u}{\sigma}\right)^{-1/\xi}$$ where \(\lambda = \Pr[ X \leq u]\).

When \(\lambda = 0\), the "conditional" distribution is equivalent to the "non conditional" distribution.

Examples

# NOT RUN {
dgpd(0.1)
rgpd(100, 1, 2, 0.2)
qgpd(seq(0.1, 0.9, 0.1), 1, 0.5, -0.2)
pgpd(12.6, 2, 0.5, 0.1)
##for non conditional quantiles
qgpd(seq(0.9, 0.99, 0.01), 1, 0.5, -0.2, lambda = 0.9)
pgpd(2.6, 2, 2.5, 0.25, lambda = 0.5)
# }