VGAM (version 0.7-5)

gumbelUC: 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).

Value

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

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 $$G(y)=\exp (-\exp [-\frac{y-\mu }{\sigma }])$$ where $-\infty0$. Its mean is $$\mu -\sigma \ast \gamma$$ and its variance is $${\sigma }^2\ast {\pi }^2/6$$ 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.

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


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)