Extreme value cumulative distribution function

`p = evcdf(x,mu,sigma)`

[p,plo,pup] = evcdf(x,mu,sigma,pcov,alpha)

[p,plo,pup] = evcdf(___,'upper')

`p = evcdf(x,mu,sigma)`

returns the cumulative
distribution function (cdf) for the type 1 extreme value distribution,
with location parameter `mu`

and scale parameter `sigma`

,
at each of the values in `x`

. `x`

, `mu`

,
and `sigma`

can be vectors, matrices, or multidimensional
arrays that all have the same size. A scalar input is expanded to
a constant array of the same size as the other inputs. The default
values for `mu`

and `sigma`

are `0`

and `1`

,
respectively.

`[p,plo,pup] = evcdf(x,mu,sigma,pcov,alpha)`

returns
confidence bounds for `p`

when the input parameters `mu`

and `sigma`

are
estimates. `pcov`

is a 2-by-2 covariance matrix of
the estimated parameters. `alpha`

has a default value
of `0.05`

, and specifies `100(1 - alpha)`

% confidence bounds. `plo`

and `pup`

are
arrays of the same size as `p`

, containing the lower
and upper confidence bounds.

`[p,plo,pup] = evcdf(___,'upper')`

returns
the complement of the type 1 extreme value distribution cdf at each
value in `x`

, using an algorithm that more accurately
computes the extreme upper tail probabilities. You can use the `'upper'`

argument
with any of the previous syntaxes.

The function `evcdf`

computes confidence bounds
for `P`

using a normal approximation to the distribution
of the estimate

$$\frac{X-\widehat{\mu}}{\widehat{\sigma}}$$

and then transforming those bounds to the scale of the output `P`

.
The computed bounds give approximately the desired confidence level
when you estimate `mu`

, `sigma`

,
and `pcov`

from large samples, but in smaller samples
other methods of computing the confidence bounds might be more accurate.

The type 1 extreme value distribution is also known as the Gumbel
distribution. The version used here is suitable for modeling minima;
the mirror image of this distribution can be used to model maxima
by negating `X`

and subtracting the resulting distribution
values from `1`

. See Extreme Value Distribution for more details. If *x* has
a Weibull distribution, then *X* = log(*x*)
has the type 1 extreme value distribution.