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)

Arguments

x

vector of quantiles.

q

vector of quantiles.

p

vector of probabilities between 0 and 1.

n

sample size. If length(n) is larger than 1, then length(n) random values are returned.

location

vector of location parameters.

scale

vector of positive scale parameters.

Details

Let \(X\) be an extreme value random variable with parameters location=\(\eta\) and scale=\(\theta\). The density function of \(X\) is given by: $$f(x; \eta, \theta) = \frac{1}{\theta} e^{-(x-\eta)/\theta} exp[-e^{-(x-\eta)/\theta}]$$ where \(-\infty < x, \eta < \infty\) and \(\theta > 0\).

The cumulative distribution function of \(X\) is given by: $$F(x; \eta, \theta) = exp[-e^{-(x-\eta)/\theta}]$$

The \(p^{th}\) quantile of \(X\) is given by: $$x_{p} = \eta - \theta log[-log(p)]$$

The mode, mean, variance, skew, and kurtosis of \(X\) are given by: $$Mode(X) = \eta$$ $$E(X) = \eta + \epsilon \theta$$ $$Var(X) = \theta^2 \pi^2 / 6$$ $$Skew(X) = \sqrt{\beta_1} = 1.139547$$ $$Kurtosis(X) = \beta_2 = 5.4$$ where \(\epsilon\) 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.

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.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

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 \(n\) 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=\(\lambda\). Then \(X = \eta - log(Y)\) has an extreme value distribution with parameters location=\(\eta\) and scale=\(1/\lambda\).

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=\(\eta\) and scale=\(\theta\), then \(Y = -X\) has a smallest extreme value distribution with parameters location=\(-\eta\) and scale=\(\theta\). The smallest extreme value distribution is related to the Weibull distribution as follows. Let \(Y\) be a Weibull random variable with parameters shape=\(\beta\) and scale=\(\alpha\). Then \(X = log(Y)\) has a smallest extreme value distribution with parameters location=\(log(\alpha)\) and scale=\(1/\beta\).

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.

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
  #[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
  #[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
  #[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
  #[1] 9.070406 7.669139 4.511481 5.903675