Extreme value inverse cumulative distribution function

`X = evinv(P,mu,sigma)`

[X,XLO,XUP] = evinv(P,mu,sigma,pcov,alpha)

`X = evinv(P,mu,sigma)`

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

and scale
parameter `sigma`

, evaluated at the values in `P`

. `P`

, `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.

`[X,XLO,XUP] = evinv(P,mu,sigma,pcov,alpha)`

produces
confidence bounds for `X`

when the input parameters `mu`

and `sigma`

are
estimates. `pcov`

is the covariance matrix of the
estimated parameters. `alpha`

is a scalar that specifies
100(1 – `alpha`

)% confidence bounds for the
estimated parameters, and has a default value of 0.05. `XLO`

and `XUP`

are
arrays of the same size as `X`

containing the lower
and upper confidence bounds.

The function `evinv`

computes confidence bounds
for `P`

using a normal approximation to the distribution
of the estimate

$$\widehat{\mu}+\widehat{\sigma}q$$

where *q* is the `P`

th quantile
from an extreme value distribution with parameters *μ
= 0* and *σ = 1*. 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`

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

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