The Generalized Extreme Value Distribution

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.

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)

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.

Value

dgev gives the density, pgev gives the distribution function, qgev gives the quantile function, and rgev generates random deviates.

References

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

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.

Note

The default value of \(\xi = 0\) means the default distribution is the Gumbel.

Currently, these functions have different argument names compared with those in the evd package.

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)
#>  [1] 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
#> [16] 0.80 0.85 0.90 0.95
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)
#> Error in h(simpleError(msg, call)): error in evaluating the argument 'x' in selecting a method for function 'plot': object 'x' not found
abline(h = 0, col = "blue", lty = 2)
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet
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")
#> Error in plot.xy(xy.coords(x, y), type = type, ...): plot.new has not been called yet
lines(x, pgev(x, loc, sigma, xi), type = "l", col = "orange")
#> Error: object 'x' not found
abline(h = (0:10)/10, lty = 2, col = "gray50")
#> Error in int_abline(a = a, b = b, h = h, v = v, untf = untf, ...): plot.new has not been called yet
 # \dontrun{}