R: The Generalized Pareto Distribution

Generalized Pareto Distribution {SpatialExtremes} | R Documentation

## 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 <= 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 = σ and shape = ξ is

G(z) = 1 - [ 1 + ξ ( x - u ) / σ ]^(-1/ξ)

for 1 + ξ (x - u) / σ > 0 and x > u, where σ > 0. If ξ = 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[ X >= x | X > u ] = 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 <= x]. Using the conditional probability definition, one have :

(1 - λ) ( 1 + ξ (x - u) /σ)_+^(-1/ξ)

where λ = Pr[X <= u].

When λ = 0, the "conditional" distribution is equivalent to the "non conditional" distribution.

### Examples

    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)

[Package SpatialExtremes version 2.0-5 Index]