Gumbel | R Documentation |
Density function, distribution function, quantile function, random
generation and raw moments for the Gumbel extreme value distribution
with parameters alpha
and scale
.
dgumbel(x, alpha, scale, log = FALSE)
pgumbel(q, alpha, scale, lower.tail = TRUE, log.p = FALSE)
qgumbel(p, alpha, scale, lower.tail = TRUE, log.p = FALSE)
rgumbel(n, alpha, scale)
mgumbel(order, alpha, scale)
mgfgumbel(t, alpha, scale, log = FALSE)
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
alpha |
location parameter. |
scale |
parameter. Must be strictly positive. |
log, log.p |
logical; if |
lower.tail |
logical; if |
order |
order of the moment. Only values |
t |
numeric vector. |
The Gumbel distribution with parameters alpha
=
\alpha
and scale
= \theta
has distribution
function:
F(x) = \exp[-\exp(-(x - \alpha)/\theta)]
for -\infty < x < \infty
, -\infty < a <
\infty
and \theta > 0
.
The mode of the distribution is in \alpha
, the mean is
\alpha + \gamma\theta
, where \gamma
=
0.57721566
is the Euler-Mascheroni constant, and the variance is
\pi^2 \theta^2/6
.
dgumbel
gives the density,
pgumbel
gives the distribution function,
qgumbel
gives the quantile function,
rgumbel
generates random deviates,
mgumbel
gives the k
th raw moment, k = 1, 2
, and
mgfgamma
gives the moment generating function in t
.
Invalid arguments will result in return value NaN
, with a warning.
Distribution also knonw as the generalized extreme value distribution Type-I.
The "distributions"
package vignette provides the
interrelations between the continuous size distributions in
actuar and the complete formulas underlying the above functions.
Vincent Goulet vincent.goulet@act.ulaval.ca
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.
dgumbel(c(-5, 0, 10, 20), 0.5, 2)
p <- (1:10)/10
pgumbel(qgumbel(p, 2, 3), 2, 3)
curve(pgumbel(x, 0.5, 2), from = -5, to = 20, col = "red")
curve(pgumbel(x, 1.0, 2), add = TRUE, col = "green")
curve(pgumbel(x, 1.5, 3), add = TRUE, col = "blue")
curve(pgumbel(x, 3.0, 4), add = TRUE, col = "cyan")
a <- 3; s <- 4
mgumbel(1, a, s) # mean
a - s * digamma(1) # same
mgumbel(2, a, s) - mgumbel(1, a, s)^2 # variance
(pi * s)^2/6 # same
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