Generalized Pareto parameter estimates
parmhat = gpfit(X)
[parmhat,parmci] = gpfit(X)
[parmhat,parmci] = gpfit(X,alpha)
[...] = gpfit(X,alpha,options)
parmhat = gpfit(X) returns maximum likelihood estimates of the parameters for the two-parameter generalized Pareto (GP) distribution given the data in X. parmhat(1) is the tail index (shape) parameter, K and parmhat(2) is the scale parameter, sigma. gpfit does not fit a threshold (location) parameter.
[...] = gpfit(X,alpha,options) specifies control parameters for the iterative algorithm used to compute ML estimates. This argument can be created by a call to statset. See statset('gpfit') for parameter names and default values.
Other functions for the generalized Pareto, such as gpcdf allow a threshold parameter, theta. However, gpfit does not estimate theta. It is assumed to be known, and subtracted from X before calling gpfit.
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, for
 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.