Generalized Pareto cumulative distribution function


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


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 k1, and the variance is not finite when k1/2. When k0, the GP has positive density for

x > theta, or, when

k < 0, 0xθσ1k.


[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

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Introduced before R2006a

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