nlogL = gevlike (params, x) returns the negative log likelihood of the data in x corresponding to the GEV distribution with (1) shape parameter k, (2) scale parameter sigma, and (3) location parameter mu given in the three-element vector params.

[nlogL, acov] = gevlike (params, x) also returns the inverse of Fisher’s information matrix, acov. If the input parameter values in params are the maximum likelihood estimates, the diagonal elements of acov are their asymptotic variances.

[…] = gevlike (params, x, freq) accepts a frequency vector, freq, of the same size as x. freq must contain non-negative integer frequencies for the corresponding elements in x. By default, or if left empty, freq = ones (size (x)).

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.

MATLAB compatibility. At exactly k = 0 the returned ACOV deviates from MATLAB’s deliberately. Both implementations agree to thirteen digits for any k != 0, and both their k != 0 expressions converge on the values returned here as k approaches 0; MATLAB’s own k = 0 branch returns something else entirely, which its k != 0 branch therefore contradicts. The Gumbel-limit expressions used here are the limits of the general ones, so ACOV is continuous at 0. This is independent of the sample size.

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.

Further information about the generalized extreme value distribution can be found at https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution

See also: gevcdf, gevinv, gevpdf, gevrnd, gevfit, gevstat

Source Code: gevlike