gumbelUC: The Gumbel Distribution

In VGAM: Vector Generalized Linear and Additive Models

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)

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.

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.

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, 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

## 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",
     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) 
## End(Not run)

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.