qrmtools (version 0.0-15)

GEV: Generalized Extreme Value Distribution

Description

Density, distribution function, quantile function and random variate generation for the generalized extreme value distribution (GEV).

Usage

dGEV(x, shape, loc = 0, scale = 1, log = FALSE)
pGEV(q, shape, loc = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qGEV(p, shape, loc = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rGEV(n, shape, loc = 0, scale = 1)

Value

dGEV() computes the density, pGEV() the distribution function, qGEV() the quantile function and rGEV() random variates of the generalized extreme value distribution.

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations.

shape

GEV shape parameter \(\xi\), a real.

loc

GEV location parameter \(\mu\), a real.

scale

GEV scale parameter \(\sigma\), a positive real.

lower.tail

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

log, log.p

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

Author

Marius Hofert

Details

The distribution function of the generalized extreme value distribution is given by $$F(x) = \cases{ \exp(-(1-\xi(x-\mu)/\sigma)^{-1/\xi}),&if $\xi\neq 0,\ 1+\xi(x-\mu)/\sigma>0$,\cr \exp(-e^{-(x-\mu)/\sigma}),&if $\xi = 0$,\cr}$$ where \(\sigma>0\).

References

McNeil, A. J., Frey, R., and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.

Examples

## Basic sanity checks
plot(pGEV(rGEV(1000, shape = 0.5), shape = 0.5)) # should be U[0,1]
curve(dGEV(x, shape = 0.5), from = -3, to = 5)