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
where q is the Pth 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.