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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 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 `k``1/2`. When `k``0`, 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.

## Version History

Introduced before R2006a