# 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 k1, and the variance is not finite when k1/2. When k0, 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