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Copula.Families.MarshallOlkin

← Copula mathematical handbook

Marshall–Olkin and Cuadras–Augé copulas #

The two-parameter Marshall–Olkin CDF is min(u^α,v^β) * u^(1-α) * v^(1-β), for α,β ∈ [0,1]. This includes the singular component and all parameter endpoints. The common-shock construction also works in every finite dimension.

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noncomputable def ProbabilityTheory.Copula.commonShock (d : ℕ) (a : Fin d → ↑unitInterval) :
Copula d

A common shock together with independent coordinate-specific shocks.

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    theorem ProbabilityTheory.Copula.isExtremeValue_commonShock (d : ℕ) (a : Fin d → ↑unitInterval) :
    (commonShock d a).IsExtremeValue
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    noncomputable def ProbabilityTheory.Copula.marshallOlkin (α β : ↑unitInterval) :
    Copula 2

    The bivariate Marshall–Olkin copula.

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      theorem ProbabilityTheory.Copula.cdf_marshallOlkin (α β : ↑unitInterval) (u : Fin 2 → ↑unitInterval) :
      (marshallOlkin α β).cdf u = min (↑(u 0) ^ ↑α) (↑(u 1) ^ ↑β) * (↑(u 0) ^ (1 - ↑α) * ↑(u 1) ^ (1 - ↑β))
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      theorem ProbabilityTheory.Copula.isExtremeValue_marshallOlkin (α β : ↑unitInterval) :
      (marshallOlkin α β).IsExtremeValue
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      noncomputable def ProbabilityTheory.Copula.cuadrasAuge (α : ↑unitInterval) :
      Copula 2

      The Cuadras–Augé family is the equal-weight Marshall–Olkin subfamily.

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        theorem ProbabilityTheory.Copula.isExtremeValue_cuadrasAuge (α : ↑unitInterval) :
        (cuadrasAuge α).IsExtremeValue
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        @[simp]
        theorem ProbabilityTheory.Copula.marshallOlkin_zero_zero :
        marshallOlkin 0 0 = independence 2
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        @[simp]
        theorem ProbabilityTheory.Copula.marshallOlkin_one_one :
        marshallOlkin 1 1 = comonotonic 2