The Extreme Value (Gumbel) Distribution
Density, distribution function, quantile function, and random generation for the (largest) extreme value distribution.
devd(x, location = 0, scale = 1) pevd(q, location = 0, scale = 1) qevd(p, location = 0, scale = 1) revd(n, location = 0, scale = 1)
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. |
Let X be an extreme value random variable with parameters
location=η and scale=θ.
The density function of X is given by:
f(x; η, θ) = \frac{1}{θ} e^{-(x-η)/θ} exp[-e^{-(x-η)/θ}]
where -∞ < x, η < ∞ and θ > 0.
The cumulative distribution function of X is given by:
F(x; η, θ) = exp[-e^{-(x-η)/θ}]
The p^{th} quantile of X is given by:
x_{p} = η - θ log[-log(p)]
The mode, mean, variance, skew, and kurtosis of X are given by:
Mode(X) = η
E(X) = η + ε θ
Var(X) = θ^2 π^2 / 6
Skew(X) = √{β_1} = 1.139547
Kurtosis(X) = β_2 = 5.4
where ε denotes Euler's constant,
which is equivalent to -digamma(1).
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.
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 n approaches infinity) of the greatest value among n independent random variables each having the same continuous distribution.
The Gumbel extreme value distribution is related to the
exponential distribution as follows.
Let Y be an exponential random variable
with parameter rate=λ. Then X = η - log(Y)
has an extreme value distribution with parameters
location=η and scale=1/λ.
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 n approaches infinity) of the smallest value among
n independent random variables each having the same continuous distribution.
If X has a largest extreme value distribution with parameters location=η and scale=θ, then
Y = -X has a smallest extreme value distribution with parameters
location=-η and scale=θ. The smallest
extreme value distribution is related to the
Weibull distribution as follows.
Let Y be a Weibull random variable with parameters
shape=β and scale=α. Then X = log(Y)
has a smallest extreme value distribution with parameters location=log(α)
and scale=1/β.
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.
Steven P. Millard (EnvStats@ProbStatInfo.com)
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.
# 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
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