VGAM (version 1.1-9)

gevUC: The Generalized Extreme Value Distribution

Description

Density, distribution function, quantile function and random generation for the generalized extreme value distribution (GEV) with location parameter location, scale parameter scale and shape parameter shape.

Usage

dgev(x, location = 0, scale = 1, shape = 0, log = FALSE,
     tolshape0 = sqrt(.Machine$double.eps))
pgev(q, location = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
qgev(p, location = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
rgev(n, location = 0, scale = 1, shape = 0)

Value

dgev gives the density,

pgev gives the distribution function,

qgev gives the quantile function, and

rgev generates random deviates.

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1 then the length is taken to be the number required.

location

the location parameter \(\mu\).

scale

the (positive) scale parameter \(\sigma\). Must consist of positive values.

shape

the shape parameter \(\xi\).

log

Logical. If log = TRUE then the logarithm of the density is returned.

lower.tail, log.p

Same meaning as in punif or qunif.

tolshape0

Positive numeric. Threshold/tolerance value for resting whether \(\xi\) is zero. If the absolute value of the estimate of \(\xi\) is less than this value then it will be assumed zero and a Gumbel distribution will be used.

Author

T. W. Yee

Details

See gev, the VGAM family function for estimating the 3 parameters by maximum likelihood estimation, for formulae and other details. Apart from n, all the above arguments may be vectors and are recyled to the appropriate length if necessary.

References

Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. London: Springer-Verlag.

See Also

gev, gevff, vglm.control.

Examples

 loc <- 2; sigma <- 1; xi <- -0.4
pgev(qgev(seq(0.05, 0.95, by = 0.05), loc, sigma, xi), loc, sigma, xi)
if (FALSE)  x <- seq(loc - 3, loc + 3, by = 0.01)
plot(x, dgev(x, loc, sigma, xi), type = "l", col = "blue", ylim = c(0, 1),
     main = "Blue is density, orange is the CDF",
     sub = "Purple are 10,...,90 percentiles", ylab = "", las = 1)
abline(h = 0, col = "blue", lty = 2)
lines(qgev(seq(0.1, 0.9, by = 0.1), loc, sigma, xi),
      dgev(qgev(seq(0.1, 0.9, by = 0.1), loc, sigma, xi), loc, sigma, xi),
      col = "purple", lty = 3, type = "h")
lines(x, pgev(x, loc, sigma, xi), type = "l", col = "orange")
abline(h = (0:10)/10, lty = 2, col = "gray50")