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# gpfit

Generalized Pareto parameter estimates

## Syntax

parmhat = gpfit(X)
[parmhat,parmci] = gpfit(X)
[parmhat,parmci] = gpfit(X,alpha)
[...] = gpfit(X,alpha,options)

## Description

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.

[parmhat,parmci] = gpfit(X) returns 95% confidence intervals for the parameter estimates.

[parmhat,parmci] = gpfit(X,alpha) returns 100(1-alpha)% confidence intervals for the parameter estimates.

[...] = 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 K1, and the variance is not finite when K1/2. When K0, the GP has positive density for

X > theta, or, when K < 0, for

$0\le \text{\hspace{0.17em}}\frac{x-\theta }{\sigma }\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}-\frac{1}{k}$