smoothtail (version 1.0)

gpd: The Generalized Pareto Distribution

Description

Density function, distribution function, quantile function and random generation for the generalized Pareto distribution (GPD) with shape parameter $\gamma$ and scale parameter $\sigma$.

Usage

dgpd(x, gam, sigma = 1) 
pgpd(q, gam, sigma = 1) 
qgpd(p, gam, sigma = 1)
rgpd(n, gam, sigma = 1)

Arguments

x, q

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

gam

Shape parameter, real number.

sigma

Scale parameter, positive real number.

Value

• dgpd gives the values of the density function, pgpd those of the distribution function, and qgpd those of the quantile function of the GPD at $x, q,$ and $p$, respectively. rgpd generates $n$ random numbers, returned as an ordered vector.

Details

The generalized Pareto distribution function (Pickands, 1975) with shape parameter $\gamma$ and scale parameter $\sigma$ is $$W_{\gamma,\sigma}(x) = 1 - {(1+\gamma x / \sigma)}_+^{-1/\gamma}.$$ If $\gamma = 0$, the distribution function is defined by continuity. The density is denoted by $w_{\gamma, \sigma}$.

References

Pickands, J. (1975). Statistical inference using extreme order statistics. Annals of Statistics, 3, 119-131.

See Also

Similar functions are provided in the R-packages evir and evd.