Generalized extreme value cumulative distribution function


p = gevcdf(x,k,sigma,mu)
p = gevcdf(x,k,sigma,mu,'upper')


p = gevcdf(x,k,sigma,mu) returns the cdf of the generalized extreme value (GEV) distribution with shape parameter k, scale parameter sigma, and location parameter, mu, evaluated at the values in x. The size of p is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

p = gevcdf(x,k,sigma,mu,'upper') returns the complement of the cdf of the GEV distribution, using an algorithm that more accurately computes the extreme upper tail probabilities.

Default values for k, sigma, and mu are 0, 1, and 0, respectively.

When k < 0, the GEV is the type III extreme value distribution. When k > 0, the GEV distribution is the type II, or Frechet, extreme value distribution. If w has a Weibull distribution as computed by the wblcdf function, then -w has a type III extreme value distribution and 1/w has a type II extreme value distribution. In the limit as k approaches 0, the GEV is the mirror image of the type I extreme value distribution as computed by the evcdf function.

The mean of the GEV distribution is not finite when k1, and the variance is not finite when k1/2. The GEV distribution has positive density only for values of X such that k*(X-mu)/sigma > -1.


[1] Embrechts, P., C. Klüppelberg, and T. Mikosch. Modelling Extremal Events for Insurance and Finance. New York: Springer, 1997.

[2] Kotz, S., and S. Nadarajah. Extreme Value Distributions: Theory and Applications. London: Imperial College Press, 2000.

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