← Copula mathematical handbook

Exact signed conditional-monotonicity classification for Clayton copulas

The strict midpoint comparison with independence excludes the opposite conditional monotonicity for each nonzero parameter branch.

theorem ProbabilityTheory.Copula.not_isCD_clayton_positive (θ : ℝ) (hθ : 0 < θ) : ¬(clayton 2 θ hθ).IsCD

Positive Clayton parameters are never conditionally decreasing.

theorem ProbabilityTheory.Copula.not_isCI_clayton_negative (θ : ℝ) (hθ : -1 ≤ θ) (hn : θ < 0) : ¬(claytonNegative θ hθ hn).IsCI

Negative Clayton parameters are never conditionally increasing.

theorem ProbabilityTheory.Copula.not_isNQD_clayton_positive (θ : ℝ) (hθ : 0 < θ) : ¬(clayton 2 θ hθ).IsNQD

Positive Clayton parameters are not negative-quadrant dependent.

theorem ProbabilityTheory.Copula.not_isPQD_clayton_negative (θ : ℝ) (hθ : -1 ≤ θ) (hn : θ < 0) : ¬(claytonNegative θ hθ hn).IsPQD

Negative Clayton parameters are not positive-quadrant dependent.