gumbelUC {VGAM}R Documentation

The Gumbel Distribution

Description

Density, distribution function, quantile function and random generation for the Gumbel distribution with location parameter location and scale parameter scale.

Usage

dgumbel(x, location=0, scale=1)
pgumbel(q, location=0, scale=1)
qgumbel(p, location=0, scale=1)
rgumbel(n, location=0, scale=1)

Arguments

x, q vector of quantiles.
p vector of probabilities.
n number of observations. Positive integer of length 1.
location the location parameter mu. This is not the mean of the Gumbel distribution (see Details below).
scale the scale parameter sigma. This is not the standard deviation of the Gumbel distribution (see Details below).

Details

The Gumbel distribution is a special case of the generalized extreme value (GEV) distribution where the shape parameter xi = 0. The latter has 3 parameters, so the Gumbel distribution has two. The Gumbel distribution function is

G(y) = exp( -exp[ - (y-mu)/sigma ] )

where -Inf<y<Inf, -Inf<mu<Inf and sigma>0. Its mean is

mu - sigma * gamma

and its variance is

sigma^2 * pi^2 / 6

where gamma is Euler's constant (which can be obtained as -digamma(1)).

See gumbel, the VGAM family function for estimating the two 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.

Value

dgumbel gives the density, pgumbel gives the distribution function, qgumbel gives the quantile function, and rgumbel generates random deviates.

Note

The VGAM family function gumbel can estimate the parameters of a Gumbel distribution using maximum likelihood estimation.

Author(s)

T. W. Yee

References

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

See Also

gumbel, egumbel, gev.

Examples

mu = 1
sigma = 2
y = rgumbel(n=100, loc=mu, scale=sigma)
mean(y)
mu - sigma * digamma(1)  # population mean
var(y)
sigma^2 * pi^2 / 6       # population variance

## Not run: 
x = seq(-2.5, 3.5, by=0.01)
loc = 0; sigma = 1
plot(x, dgumbel(x, loc, sigma), type="l", col="blue", ylim=c(0,1),
     main="Blue is density, red is cumulative distribution function",
     sub="Purple are 5,10,...,95 percentiles", ylab="", las=1)
abline(h=0, col="blue", lty=2)
lines(qgumbel(seq(0.05,0.95,by=0.05), loc, sigma),
      dgumbel(qgumbel(seq(0.05,0.95,by=0.05), loc, sigma), loc, sigma),
      col="purple", lty=3, type="h")
lines(x, pgumbel(x, loc, sigma), type="l", col="red")
abline(h=0, lty=2)
## End(Not run)

[Package VGAM version 0.7-7 Index]