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Quadrant dependence and conditional increasingness for the Clayton family

theorem ProbabilityTheory.Copula.cdf_clayton_two_pos (θ : ℝ) (hθ : 0 < θ) (u v : ↑unitInterval) (hu : 0 < ↑u) (hv : 0 < ↑v) : (clayton 2 θ hθ).cdf ![u, v] = (↑u ^ (-θ) + ↑v ^ (-θ) - 1) ^ (-1 / θ)

The bivariate Clayton CDF for positive parameters at positive coordinates.

theorem ProbabilityTheory.Copula.isPQD_clayton_positive (θ : ℝ) (hθ : 0 < θ) : (clayton 2 θ hθ).IsPQD

Positive Clayton parameters are positively quadrant dependent.

theorem ProbabilityTheory.Copula.isNQD_clayton_negative (θ : ℝ) (hθ : -1 ≤ θ) (hn : θ < 0) : (claytonNegative θ hθ hn).IsNQD

Every admissible negative bivariate Clayton copula is negatively quadrant dependent.

theorem ProbabilityTheory.Copula.isSI_clayton_positive (θ : ℝ) (hθ : 0 < θ) : (clayton 2 θ hθ).IsSI

Every positive bivariate Clayton copula is stochastically increasing in the first coordinate.

theorem ProbabilityTheory.Copula.isCI_clayton_positive (θ : ℝ) (hθ : 0 < θ) : (clayton 2 θ hθ).IsCI

Every positive bivariate Clayton copula is conditionally increasing in both directions.