GPD function

The generalised Pareto distribution

Density, distribution function, quantile function and random generation for the Generalised Pareto Distribution (GPD).

dgpd(x, gamma, mu = 0, sigma, log = FALSE)
pgpd(x, gamma, mu = 0, sigma, lower.tail = TRUE, log.p = FALSE)
qgpd(p, gamma, mu = 0, sigma, lower.tail = TRUE, log.p = FALSE)
rgpd(n, gamma, mu = 0, sigma)

Arguments

• x: Vector of quantiles.
• p: Vector of probabilities.
• n: Number of observations.
• gamma: The $\gamma$ parameter of the GPD, a real number.
• mu: The $\mu$ parameter of the GPD, a strictly positive number. Default is 0.
• sigma: The $\sigma$ parameter of the GPD, a strictly positive number.
• log: Logical indicating if the densities are given as $\log(f)$, default is FALSE.
• lower.tail: Logical indicating if the probabilities are of the form $P(X\le x)$ (TRUE) or $P(X>x)$ (FALSE). Default is TRUE.
• log.p: Logical indicating if the probabilities are given as $\log(p)$, default is FALSE.

Details

The Cumulative Distribution Function (CDF) of the GPD for $\gamma \neq 0$ is equal to $F(x) = 1-(1+\gamma (x-\mu)/\sigma)^{-1/\gamma}$ for all $x \ge \mu$ and $F(x)=0$ otherwise. When $\gamma=0$, the CDF is given by $F(x) = 1-\exp((x-\mu)/\sigma)$ for all $x \ge \mu$ and $F(x)=0$ otherwise.

Returns

dgpd gives the density function evaluated in $x$, pgpd the CDF evaluated in $x$ and qgpd the quantile function evaluated in $p$. The length of the result is equal to the length of $x$ or $p$.

rgpd returns a random sample of length $n$.

References

Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.

Author(s)

Tom Reynkens.

See Also

tGPD, Pareto, EPD, Distributions

Examples

# Plot of the PDF
x <- seq(0, 10, 0.01)
plot(x, dgpd(x, gamma=1/2, sigma=5), xlab="x", ylab="PDF", type="l")

# Plot of the CDF
x <- seq(0, 10, 0.01)
plot(x, pgpd(x, gamma=1/2, sigma=5), xlab="x", ylab="CDF", type="l")