gumbelUC {VGAM} | R Documentation |
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 |
location |
the location parameter |
scale |
the scale parameter |
log |
Logical.
If |
lower.tail , log.p |
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)