gumbelUC | R Documentation |
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 |
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 |
lower.tail, log.p |
Same meaning as in |
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", 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)
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