# Documentation

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