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 `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`

.

[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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