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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)
vector of quantiles.
vector of probabilities.
number of observations.
If length(n) > 1
then the length is taken to be the number required.
the location parameter
the scale parameter
Logical.
If log = TRUE
then the logarithm of the density is returned.
dgumbel
gives the density,
pgumbel
gives the distribution function,
qgumbel
gives the quantile function, and
rgumbel
generates random deviates.
The Gumbel distribution is a special case of the
generalized extreme value (GEV) distribution where
the shape parameter -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.
Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. London: Springer-Verlag.
# NOT RUN {
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)
# }
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