Prove2Me

Claims deviance, premium relativity and model comparison: gmGammaVariance_nonneg

Proved
ActuarialValuation.gmGammaVariance_nonneg

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

actuarial-scienceformal-verificationinsurance-glm-pricing

Gamma severity variance is nonnegative at positive shape. This original derived actuarial relation uses the parameter restrictions, endpoints and units stated in the separate formal theorem or definition.

Mathematical relation:

DPoi=2[ylog⁡(yμ)−(y−μ)]D_{\text{Poi}}=2\left[y\log\left(\frac{y}{\mu}\right)-(y-\mu)\right]
Preamble
import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Definitions.Def_actuarial_gmGammaVariance
Formal statement
namespace ActuarialValuation

theorem gmGammaVariance_nonneg (mean shape : ℝ) (hs : 0<shape) : 0≤gmGammaVariance mean shape := by sorry

end ActuarialValuation
Source
Original derived actuarial mathematics, published source page 373. J. A. Nelder and R. W. M. Wedderburn (1972), Generalized Linear Models, JRSS A 135:370–384, https://doi.org/10.2307/2344614; actuarial practice in casualty generalized linear models https://www.casact.org/sites/default/files/2021-03/05-Goldburd-Khare-Tevet.pdf. Parent topic: Positive insurance exposure offsets, Poisson log-link likelihood and score, Fisher information, deviance and multiplicative premium relativities. 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://doi.org/10.2307/2344614
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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