# gpcdf

Generalized Pareto cumulative distribution function

## Syntax

`p = gpcdf(x,k,sigma,theta)p = gpcdf(x,k,sigma,theta,'upper')`

## Description

`p = gpcdf(x,k,sigma,theta)` returns the cdf of the generalized Pareto (GP) distribution with the tail index (shape) parameter `k`, scale parameter `sigma`, and threshold (location) parameter, `theta`, evaluated at the values in `x`. The size of `p` is the common size of the input arguments. A scalar input functions as a constant matrix of the same size as the other inputs.

`p = gpcdf(x,k,sigma,theta,'upper')` returns the complement of the cdf of the generalized Pareto (GP) distribution, using an algorithm that more accurately computes the extreme upper tail probabilities.

Default values for `k`, `sigma`, and `theta` are 0, 1, and 0, respectively.

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.

## See Also

#### Introduced before R2006a

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