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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 k1, and the variance is not finite when k1/2. The GEV distribution has positive density only for values of X such that k*(X-mu)/sigma > -1.

References

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

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

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