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Generalized Pareto Distribution
Definition
The probability density function for the generalized Pareto distribution with shape parameter k ≠ 0, scale parameter σ, and threshold parameter θ, is

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for θ < x, when k > 0, or for θ < x < –σ/k when k < 0.
In the limit for k = 0, the density is

无

for θ < x.
If k = 0 and θ = 0, the generalized Pareto distribution is equivalent to the exponential distribution. If k > 0 and θ = σ/k, the generalized Pareto distribution is equivalent to the Pareto distribution.