# gevcdf

Generalized extreme value cumulative distribution function

## Syntax

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

## Description

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

## References

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

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