Gumbel Distribution

The Gumbel distribution, also known as the Type I extreme value distribution or log-Weibull distribution, arises as the limiting distribution of properly normalized maxima or minima of independent, identically distributed random variables. It is one of the three possible limit distributions in the Fisher-Tippett theorem and is commonly applied in fields such as hydrology, meteorology, and material science.

Definition

The Gumbel distribution has a location parameter a and a scale parameter b. For modeling maxima (standard form), the probability density function (PDF) is:

f(x)=1be−(x−a)/be−e−(x−a)/b

The corresponding cumulative distribution function (CDF) is:

F(x)=e−e−(x−a)/b

For modeling minima (reverse form), the PDF is:

f(x)=1be(x−a)/be−e(x−a)/b

With CDF:

F(x)=1−e−e(x−a)/b

The domain is all real numbers, x∈R, with scale parameter b>0. The quantile function for the maximum form is:

Q(p)=a−bln⁡(−ln⁡(p))

Applications

The Gumbel distribution finds applications in several fields:

  • In hydrology, the distribution is used for predicting annual flood levels and modeling rainfall maxima.

  • Meteorologists apply the distribution to analyze extreme temperatures and wind speeds.

  • The distribution is valuable in materials science for analyzing material strength and predicting time-to-failure.

  • Risk analysts use the distribution to model extreme insurance losses and rare events.

  • Environmental scientists employ the distribution to model extreme events related to climate change.

Properties

Statistical Properties (Maximum Form)
PropertyValue
Meana+bγ, where γ is Euler's constant
Mediana−bln⁡(ln⁡(2))
Modea
Varianceπ26b2≈1.6449b2
Skewness126π3ζ(3)≈1.1396
Excess Kurtosis125
Entropyln⁡(b)+γ+1

The Gumbel distribution has several notable properties:

  • The distribution is closed under location-scale transformation.

Relationships to Other Distributions

  • The Gumbel distribution is the Type I extreme value distribution. The Type II extreme value distribution is the Fréchet or Inverse Weibull distribution. The Type III extreme value distribution is the Weibull distribution.

  • If X follows a Gumbel distribution, then -X follows a reverse Gumbel distribution.

  • It arises as the limiting distribution of the maximum of independent identically distributed random variables with exponential-like tails.

The GumbelDistribution class

The Gumbel distribution is implemented by the GumbelDistribution class. It has one constructor that takes three arguments. The first argument is the location parameter, and corresponds to the mode of the probability density function. The second argument is the scale parameter. The last, optional argument specifies whether the distribution should model the smallest or the largest extreme value. The default is to model the largest.

The following constructs the same Gumbel distribution with mode 6.8 and scale parameter 4.1:

C# 
var gumbel = new GumbelDistribution(6.8, 4.1);

The GumbelDistribution class has two specific properties, LocationParameter and ScaleParameter, which return the location parameter (mode) and scale parameter of the distribution.

GumbelDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The second and third arguments are the location and scale parameters of the distribution.

C# 
var random = new Pcg32();
double sample = GumbelDistribution.Sample(random, 6.8, 4.1);

The above example uses the Pcg32 class to generate uniform random numbers.

References

For more information on the Gumbel distribution, refer to the following sources:

  • Wikipedia: Gumbel distribution
  • MathWorld: Gumbel Distribution

References

For more information on the Gumbel distribution, refer to the following sources:

  • Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 2", Chapter 22. Wiley Series in Probability and Statistics.

  • Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 19. Wiley Series in Probability and Statistics.

See Also