← Copula mathematical handbook

The tail-dependence coefficient of the Student-t copula

For the bivariate Student-t copula C_{ν,r} with ν > 0 degrees of freedom and correlation r ∈ (−1, 1],

λ_L = λ_U = 2 t_{ν+1}(−√((ν+1)(1−r)/(1+r)))

where t_{ν+1} is the Student-t distribution function with ν + 1 degrees of freedom (Embrechts–McNeil–Straumann 2002, Demarta–McNeil 2005). In particular the t copula is tail dependent for every r > −1, unlike the Gaussian copula (hasLowerTailDependence_bivariateGaussian). For r = −1 the copula is countermonotonic and λ = 0.

Proof

Copula.Elliptical.StudentTTail.TailDependence gives the angular form λ = ∫_a^{π/2} cos^ν / ∫_0^{π/2} cos^ν with a = arccos(r)/2. With n = ν + 1, the tan substitution of Copula.Families.StudentT.Distribution gives t_n(−√n tan a) = ∫_a^{π/2} cos^{n−1} / (2 ∫_0^{π/2} cos^{n−1}), and the half-angle formula tan(arccos(r)/2) = √((1−r)/(1+r)) finishes the computation.

Main results

• studentTTailCoeff_eq_studentTCDF: the closed form of the coefficient.
• hasLowerTailDependence_studentT_closedForm, hasUpperTailDependence_studentT_closedForm.
• studentT_not_tailIndependent: λ_L ≠ 0 for r > −1.

References

• P. Embrechts, A. McNeil, D. Straumann, Correlation and dependence in risk management: properties and pitfalls, CUP 2002.
• S. Demarta, A. McNeil, The t copula and related copulas, Int. Stat. Rev. 73 (2005).
• H. Hult, F. Lindskog, Multivariate extremes, aggregation and dependence in elliptical distributions, Adv. Appl. Probab. 34 (2002).

theorem ProbabilityTheory.Copula.tan_arccos_div_two {r : ℝ} (hr : r ∈ Set.Ioc (-1) 1) : Real.tan (Real.arccos r / 2) = √((1 - r) / (1 + r))

The half-angle formula tan(arccos(r)/2) = √((1−r)/(1+r)) for r ∈ (−1, 1].

theorem ProbabilityTheory.Copula.studentTTailCoeff_eq_studentTCDF {ν r : ℝ} (hν : 0 < ν) (hr : r ∈ Set.Ioc (-1) 1) : studentTTailCoeff ν r = 2 * studentTCDF (ν + 1) (-√((ν + 1) * (1 - r) / (1 + r)))

Closed form of the Student-t tail-dependence coefficient: λ(ν, r) = 2 t_{ν+1}(−√((ν+1)(1−r)/(1+r))) for r ∈ (−1, 1].

theorem ProbabilityTheory.Copula.hasLowerTailDependence_studentT_closedForm {r : ℝ} (hr : r ∈ Set.Ioc (-1) 1) {ν : ℝ} (hν : 0 < ν) : (studentT (corrMatrix r) ⋯ ⋯ ν hν).HasLowerTailDependence (2 * studentTCDF (ν + 1) (-√((ν + 1) * (1 - r) / (1 + r))))

Lower tail dependence of the Student-t copula (Embrechts–McNeil–Straumann 2002): λ_L = 2 t_{ν+1}(−√((ν+1)(1−r)/(1+r))) for r ∈ (−1, 1].

theorem ProbabilityTheory.Copula.hasUpperTailDependence_studentT_closedForm {r : ℝ} (hr : r ∈ Set.Ioc (-1) 1) {ν : ℝ} (hν : 0 < ν) : (studentT (corrMatrix r) ⋯ ⋯ ν hν).HasUpperTailDependence (2 * studentTCDF (ν + 1) (-√((ν + 1) * (1 - r) / (1 + r))))

Upper tail dependence of the Student-t copula: λ_U = 2 t_{ν+1}(−√((ν+1)(1−r)/(1+r))) for r ∈ (−1, 1].

theorem ProbabilityTheory.Copula.studentT_not_hasLowerTailDependence_zero {r : ℝ} (hr : r ∈ Set.Ioc (-1) 1) {ν : ℝ} (hν : 0 < ν) : ¬(studentT (corrMatrix r) ⋯ ⋯ ν hν).HasLowerTailDependence 0

The Student-t copula is not lower tail independent for r > −1 (contrast with the Gaussian copula, hasLowerTailDependence_bivariateGaussian).