# 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 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 k1, and the variance is not finite when k1/2. When k0, the GP has positive density for

k > theta, or, when k < 0, for

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

## 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.