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# gpcdf

Generalized Pareto cumulative distribution function

## Syntax

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

## Description

p = gpcdf(x,x,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 the Pareto distribution. 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, $0\le \text{\hspace{0.17em}}\frac{x-\theta }{\sigma }\text{\hspace{0.17em}}\le \text{\hspace{0.17em}}-\frac{1}{k}$.