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Copula.Rank.Blest

← Copula mathematical handbook

Blest's rank correlation and its symmetrization #

The directional coefficient weights discrepancies in the top ranks. Averaging it with its transpose gives the symmetrized coefficient of Genest and Plante. These are population copula functionals, not finite-sample estimators.

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noncomputable def ProbabilityTheory.Copula.blestNu (C : Copula 2) :
ℝ

Blest's directional population rank correlation.

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    noncomputable def ProbabilityTheory.Copula.symmetrizedBlest (C : Copula 2) :
    ℝ

    Symmetrized Blest rank correlation: the average of both coordinate directions.

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      @[simp]
      theorem ProbabilityTheory.Copula.symmetrizedBlest_transpose (C : Copula 2) :
      C.transpose.symmetrizedBlest = C.symmetrizedBlest
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      theorem ProbabilityTheory.Copula.symmetrizedBlest_eq_integral (C : Copula 2) :
      C.symmetrizedBlest = -4 + 6 * ∫ (x : Fin 2 → ↑unitInterval), ↑(x 0) * ↑(x 1) * (4 - ↑(x 0) - ↑(x 1)) ∂C.toMeasure

      The symmetric mixed-moment formula fixes the normalization independently of the directional representation.

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      @[simp]
      theorem ProbabilityTheory.Copula.blestNu_independence :
      (independence 2).blestNu = 0
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      @[simp]
      theorem ProbabilityTheory.Copula.blestNu_comonotonic :
      (comonotonic 2).blestNu = 1
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      @[simp]
      theorem ProbabilityTheory.Copula.blestNu_countermonotonic :
      countermonotonic.blestNu = -1
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      @[simp]
      theorem ProbabilityTheory.Copula.symmetrizedBlest_independence :
      (independence 2).symmetrizedBlest = 0
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      @[simp]
      theorem ProbabilityTheory.Copula.symmetrizedBlest_comonotonic :
      (comonotonic 2).symmetrizedBlest = 1
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      @[simp]
      theorem ProbabilityTheory.Copula.symmetrizedBlest_countermonotonic :
      countermonotonic.symmetrizedBlest = -1
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      theorem ProbabilityTheory.Copula.blestNu_mix (C D : Copula 2) (a : ↑unitInterval) :
      (C.mix D a).blestNu = ↑a * C.blestNu + (1 - ↑a) * D.blestNu
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      theorem ProbabilityTheory.Copula.symmetrizedBlest_mix (C D : Copula 2) (a : ↑unitInterval) :
      (C.mix D a).symmetrizedBlest = ↑a * C.symmetrizedBlest + (1 - ↑a) * D.symmetrizedBlest