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gplike

Generalized Pareto negative loglikelihood

Syntax

nlogL = gplike(params,data)
[nlogL,acov] = gplike(params,data)

Description

nlogL = gplike(params,data) returns the negative of the loglikelihood nlogL for the two-parameter generalized Pareto (GP) distribution, evaluated at parameters params. params(1) is the tail index (shape) parameter, k, and params(2) is the scale parameter. gplike does not allow a threshold (location) parameter.

[nlogL,acov] = gplike(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 and theta = 0, the GP is equivalent to the exponential distribution. When k > 0 and theta = sigma/k, the GP is equivalent to a Pareto distribution with a scale parameter equal to sigma/k and a shape parameter equal to 1/k. The mean of the GP is not finite when k ≥ 1, and the variance is not finite when k1/2. When k0, the GP has positive density for

x > theta, or, when

k < 0, 0xθσ1k.

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.

Introduced before R2006a