# gplike

Generalized Pareto negative loglikelihood

## Syntax

```nlogL = gplike(params,data) [nlogL,acov] = gplike(params,data) ```

## Description

`nlogL = gplike(params,data)` returns the negative of the loglikelihood `nlogL` for the two-parameter generalized Pareto (GP) distribution, evaluated at parameters `params`. `params(1)` is the tail index (shape) parameter, `k`, and `params(2)` is the scale parameter. `gplike` does not allow a threshold (location) parameter.

`[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 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

`x > theta`, or, when

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

## 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