# gprnd

Generalized Pareto random numbers

## Syntax

```r = gprnd(k,sigma,theta) r = gprnd(k,sigma,theta,m,n,...) R = gprnd(K,sigma,theta,[m,n,...]) ```

## Description

`r = gprnd(k,sigma,theta)` returns an array of random numbers chosen from the generalized Pareto (GP) distribution with tail index (shape) parameter `k`, scale parameter `sigma`, and threshold (location) parameter, `theta`. The size of `r` is the common size of the input arguments if all are arrays. If any parameter is a scalar, the size of `r` is the size of the other parameters.

`r = gprnd(k,sigma,theta,m,n,...)` or ```R = gprnd(K,sigma,theta,[m,n,...])``` generates an `m`-by-`n`-by-... array. The `k`, `sigma`, `theta` parameters can each be scalars or arrays of the same size as `r`.

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

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

## 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.

## Extended Capabilities

Introduced before R2006a