Generalized Pareto negative log-likelihood
nlogL = gplike(params,data)
[nlogL,ACOV] = gplike(params,data)
nlogL = gplike(params,data) returns the negative of the log-likelihood nlogL for the two-parameter generalized Pareto (GP) distribution, evaluated at parameters params. params(1) is the tail index (shape) parameter, K, params(2) is the scale parameter, sigma, and params(3) is the threshold (location) parameter, mu.
[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 the Pareto distribution. The mean of the GP is not finite when K ≥ 1, and the variance is not finite when K ≥ 1/2. When K ≥ 0, the GP has positive density for
X > theta, or, when
K < 0, .
 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.