gumbelUC: The Gumbel Distribution
In VGAM: Vector Generalized Linear and Additive Models

Description Usage Arguments Details Value Note Author(s) References See Also Examples

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

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)

`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`.

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.

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

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

T. W. Yee

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

gumbel, gumbelff, gev, dgompertz.

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

## 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)