Exact lower and upper tail dependence of Ali–Mikhail–Haq copulas #
theorem
ProbabilityTheory.Copula.hasLowerTailDependence_amh_lt_one
(θ : ℝ)
(hmin : -1 ≤ θ)
(hmax : θ ≤ 1)
(hθ : θ < 1)
:
(amh θ hmin hmax).HasLowerTailDependence 0
AMH has zero lower-tail dependence for every parameter below one.
theorem
ProbabilityTheory.Copula.hasLowerTailDependence_amh_one :
(amh 1 ⋯ ⋯).HasLowerTailDependence (1 / 2)
The θ=1 AMH endpoint is Clayton(1), with lower-tail coefficient 1/2.
theorem
ProbabilityTheory.Copula.hasUpperTailDependence_amh
(θ : ℝ)
(hmin : -1 ≤ θ)
(hmax : θ ≤ 1)
:
(amh θ hmin hmax).HasUpperTailDependence 0
Every admissible AMH parameter has zero upper-tail dependence.