# gevlike

Generalized extreme value negative log-likelihood

## Syntax

```nlogL = gevlike(params,data) [nlogL,ACOV] = gevlike(params,data) ```

## Description

`nlogL = gevlike(params,data)` returns the negative of the log-likelihood `nlogL` for the generalized extreme value (GEV) distribution, evaluated at parameters `params`. `params(1)` is the shape parameter, `k`, `params(2)` is the scale parameter, `sigma`, and `params(3)` is the location parameter, `mu`.

`[nlogL,ACOV] = gevlike(params,data)` 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. `ACOV` is based on the observed Fisher's information, not the expected information.

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 `wbllike` 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 `evlike` function.

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`.

## References

 Embrechts, P., C. Klüppelberg, and T. Mikosch. Modelling Extremal Events for Insurance and Finance. New York: Springer, 1997.

 Kotz, S., and S. Nadarajah.Extreme Value Distributions: Theory and Applications. London: Imperial College Press, 2000.