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# evcdf

Extreme value cumulative distribution function

## Syntax

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

## Description

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

$\frac{X-\stackrel{^}{\mu }}{\stackrel{^}{\sigma }}$

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.