Documentation

# gpstat

Generalized Pareto mean and variance

## Syntax

```[m,v] = gpstat(k,sigma,theta) ```

## Description

`[m,v] = gpstat(k,sigma,theta)` returns the mean of and variance for the generalized Pareto (GP) distribution with the tail index (shape) parameter `k`, scale parameter `sigma`, and threshold (location) parameter, `theta`.

The default value for `theta` is 0.

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.