← Copula mathematical handbook

The outer-power transformation of an Archimedean generator

noncomputable def ProbabilityTheory.Copula.BivariateGenerator.outerPower (g : BivariateGenerator) (θ : ℝ) (hθ : 1 ≤ θ) : BivariateGenerator

The outer-power generator t ↦ ψ(t^(1/θ)), for θ ≥ 1. Applied to exponential and Clayton generators this gives Gumbel and BB1.

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@[simp]

theorem ProbabilityTheory.Copula.BivariateGenerator.outerPower_one (g : BivariateGenerator) : g.outerPower 1 ⋯ = g

noncomputable def ProbabilityTheory.Copula.BivariateGenerator.innerPower (g : BivariateGenerator) (θ : ℝ) (hθ : 1 ≤ θ) : BivariateGenerator

Raising the inverse generator to a power at least one preserves bivariate admissibility, including generators with a finite zero.

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@[simp]

theorem ProbabilityTheory.Copula.BivariateGenerator.innerPower_one (g : BivariateGenerator) : g.innerPower 1 ⋯ = g