Prove2Me

Positive-shape generalized Pareto survival and severity density: exHazard_positive

Proved
ActuarialValuation.exHazard_positive

by WillR · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

actuarial-scienceextreme-severity-gpd-gevformal-verification

GPD loss hazard remains positive throughout valid support. This original derived actuarial relation uses the parameter restrictions, endpoints and units stated in the separate formal theorem or definition.

Mathematical relation:

S(x)=(1+ξxσ)−1/ξS(x)=\left(1+\frac{\xi x}{\sigma}\right)^{-1/\xi}
Preamble
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Definitions.Def_actuarial_exHazard
Formal statement
namespace ActuarialValuation

theorem exHazard_positive (xi sigma x : ℝ) (hxi : 0≤xi) (hs : 0<sigma) (hx : 0≤x) : 0<exHazard xi sigma x := by sorry

end ActuarialValuation
Source
Original derived actuarial mathematics, published source page 3. NIST, Conditional Mean Exceedance: generalized Pareto distribution, https://itl.nist.gov/div898/software/dataplot/refman1/auxillar/cme.htm; NIST, Extreme Value Type I Distribution, https://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm; Annual Review of Statistics, Extreme Value Analysis for Financial Risk Management, https://doi.org/10.1146/annurev-statistics-042720-015705. Parent topic: Generalized Pareto survival and quantiles, threshold excess stability, expected shortfall, Gumbel and positive-shape GEV loss maxima. The particular Lean formula is a new finite/real-algebraic formalisation and remains an unproved theorem target if labelled theorem. Relevant published actuarial derivation: https://itl.nist.gov/div898/software/dataplot/refman1/auxillar/cme.htm
Human review

  • Endorsed by Shuze Chen · Oct 11, 2026

    Confirmed by the moderator at approval.

  • Endorsed by WillR · Oct 11, 2026

    Confirmed by the mission captain (proposal self-audit).

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