Generalized Pareto Distribution
X ~ GPD(μ, σ, ξ)
Extreme-Value / Heavy-Tailed Family • Published July 22, 2026
The distribution behind peaks-over-threshold extreme value theory: history from the Pickands–Balkema–de Haan theorem, a shape parameter ξ that continuously interpolates between the already-published Pareto's heavy unbounded tail (ξ>0), the already-published Exponential's light unbounded tail (ξ=0), and a hard finite upper bound (ξ<0), confirmed end-to-end by simulating exceedances of the already-published Pareto over a high threshold and fitting a GPD to them, no closed-form maximum-likelihood system, and simulation via closed-form inverse-transform sampling.
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About This Reference Sheet
The distribution behind peaks-over-threshold extreme value theory: history from the Pickands–Balkema–de Haan theorem, a shape parameter ξ that continuously interpolates between the already-published Pareto's heavy unbounded tail (ξ>0), the already-published Exponential's light unbounded tail (ξ=0), and a hard finite upper bound (ξ<0), confirmed end-to-end by simulating exceedances of the already-published Pareto over a high threshold and fitting a GPD to them, no closed-form maximum-likelihood system, and simulation via closed-form inverse-transform sampling.
Support
x ≥ μ (bounded above if ξ < 0)
Parameters
μ ∈ ℝ (threshold), σ > 0 (scale), ξ ∈ ℝ (shape)