GumbelDistribution Class

In probability theory and statistics, the Gumbel distribution is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. Such a distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur.

The potential applicability of the Gumbel distribution to represent the distribution of maxima relates to extreme value theory which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type.

The Gumbel distribution is a particular case of the generalized extreme value distribution (also known as the Fisher-Tippett distribution). It is also known as the log-Weibull distribution and the double exponential distribution (a term that is alternatively sometimes used to refer to the Laplace distribution). It is related to the Gompertz distribution[citation needed]: when its density is first reflected about the origin and then restricted to the positive half line, a Gompertz function is obtained.

In the latent variable formulation of the multinomial logit model — common in discrete choice theory — the errors of the latent variables follow a Gumbel distribution. This is useful because the difference of two Gumbel-distributed random variables has a logistic distribution.

The Gumbel distribution is named after Emil Julius Gumbel (1891–1966), based on his original papers describing the distribution.

References:

• Wikipedia, The Free Encyclopedia. Gumbel distribution. Available on: http://en.wikipedia.org/wiki/Gumbel_distribution

The following example shows how to create and test the main characteristics of an Gumbel distribution given its location and scale parameters:

var gumbel = new GumbelDistribution(location: 4.795, scale: 1 / 0.392);

double mean = gumbel.Mean;     // 6.2674889410753387
double median = gumbel.Median; // 5.7299819402593481
double mode = gumbel.Mode;     // 4.7949999999999999
double var = gumbel.Variance;  // 10.704745853604138

double cdf = gumbel.DistributionFunction(x: 3.4); // 0.17767760424788051
double pdf = gumbel.ProbabilityDensityFunction(x: 3.4); // 0.12033954114322486
double lpdf = gumbel.LogProbabilityDensityFunction(x: 3.4); // -2.1174380222001519

double ccdf = gumbel.ComplementaryDistributionFunction(x: 3.4); // 0.82232239575211952
double icdf = gumbel.InverseDistributionFunction(p: cdf); // 3.3999999904866245

double hf = gumbel.HazardFunction(x: 1.4); // 0.03449691276402958
double chf = gumbel.CumulativeHazardFunction(x: 1.4); // 0.022988793482259906

string str = gumbel.ToString(CultureInfo.InvariantCulture); // Gumbel(x; μ = 4.795, β = 2.55)

Reference