VGAM (version 1.1-9)

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, log = FALSE)
pgumbel(q, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
qgumbel(p, location = 0, scale = 1, lower.tail = TRUE, log.p = FALSE)
rgumbel(n, location = 0, scale = 1)

Value

dgumbel gives the density,

pgumbel gives the distribution function,

qgumbel gives the quantile function, and

rgumbel 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 \(\mu\). 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).

log

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

lower.tail, log.p

Same meaning as in punif or qunif.

Author

T. W. Yee

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 $$G(y) = \exp \left( - \exp \left[ - \frac{y-\mu}{\sigma} \right] \right) $$ where \(-\infty<y<\infty\), \(-\infty<\mu<\infty\) 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.

References

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

See Also

gumbel, gumbelff, gev, dgompertz.

Examples

mu <- 1; sigma <- 2;
y <- rgumbel(n = 100, loc = mu, scale = sigma)
c(mean(y), mu - sigma * digamma(1))  # Sample and population means
c(var(y), sigma^2 * pi^2 / 6)  # Sample and population variances

if (FALSE)  x <- seq(-2.5, 3.5, by = 0.01)
loc <- 0; sigma <- 1
plot(x, dgumbel(x, loc, sigma), type = "l", col = "blue",
     main = "Blue is density, red is the CDF", ylim = c(0, 1),
     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)