EVD {EnvStats} | R Documentation |
The Extreme Value (Gumbel) Distribution
Description
Density, distribution function, quantile function, and random generation for the (largest) extreme value distribution.
Usage
devd(x, location = 0, scale = 1)
pevd(q, location = 0, scale = 1)
qevd(p, location = 0, scale = 1)
revd(n, location = 0, scale = 1)
Arguments
x |
vector of quantiles. |
q |
vector of quantiles. |
p |
vector of probabilities between 0 and 1. |
n |
sample size. If |
location |
vector of location parameters. |
scale |
vector of positive scale parameters. |
Details
Let be an extreme value random variable with parameters
location=
and
scale=
.
The density function of
is given by:
where and
.
The cumulative distribution function of is given by:
The quantile of
is given by:
The mode, mean, variance, skew, and kurtosis of are given by:
where denotes Euler's constant,
which is equivalent to
-digamma(1)
.
Value
density (devd
), probability (pevd
), quantile (qevd
), or
random sample (revd
) for the extreme value distribution with
location parameter(s) determined by location
and scale
parameter(s) determined by scale
.
Note
There are three families of extreme value distributions. The one
described here is the Type I, also called the Gumbel extreme value
distribution or simply Gumbel distribution. The name
“extreme value” comes from the fact that this distribution is
the limiting distribution (as approaches infinity) of the
greatest value among
independent random variables each
having the same continuous distribution.
The Gumbel extreme value distribution is related to the
exponential distribution as follows.
Let be an exponential random variable
with parameter
rate=
. Then
has an extreme value distribution with parameters
location=
and
scale=
.
The distribution described above and used by devd
, pevd
,
qevd
, and revd
is the largest extreme value
distribution. The smallest extreme value distribution is the limiting
distribution (as approaches infinity) of the smallest value among
independent random variables each having the same continuous distribution.
If
has a largest extreme value distribution with parameters
location=
and
scale=
, then
has a smallest extreme value distribution with parameters
location=
and
scale=
. The smallest
extreme value distribution is related to the
Weibull distribution as follows.
Let
be a Weibull random variable with parameters
shape=
and
scale=
. Then
has a smallest extreme value distribution with parameters
location=
and
scale=
.
The extreme value distribution has been used extensively to model the distribution of streamflow, flooding, rainfall, temperature, wind speed, and other meteorological variables, as well as material strength and life data.
Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
References
Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1995). Continuous Univariate Distributions, Volume 2. Second Edition. John Wiley and Sons, New York.
See Also
eevd
, GEVD
,
Probability Distributions and Random Numbers.
Examples
# Density of an extreme value distribution with location=0, scale=1,
# evaluated at 0.5:
devd(.5)
#[1] 0.3307043
#----------
# The cdf of an extreme value distribution with location=1, scale=2,
# evaluated at 0.5:
pevd(.5, 1, 2)
#[1] 0.2769203
#----------
# The 25'th percentile of an extreme value distribution with
# location=-2, scale=0.5:
qevd(.25, -2, 0.5)
#[1] -2.163317
#----------
# Random sample of 4 observations from an extreme value distribution with
# location=5, scale=2.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(20)
revd(4, 5, 2)
#[1] 9.070406 7.669139 4.511481 5.903675