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Tail dependence 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], the lower and upper tail-dependence coefficients coincide and equal

λ = ∫_{a}^{π/2} cos^ν θ dθ / ∫_0^{π/2} cos^ν θ dθ, a = arccos(r) / 2

(Hult–Lindskog 2002, angular form). In particular λ > 0 for every r > −1: in contrast to the Gaussian copula (hasLowerTailDependence_bivariateGaussian), the t copula is tail dependent even for negative correlations. The closed form λ = 2 t_{ν+1}(−√((ν+1)(1−r)/(1+r))) of Embrechts, McNeil and Straumann is derived in Copula.Elliptical.StudentTTail.

Proof

C(F(x), F(x)) / F(x) = P(X ≤ x, Y ≤ x) / P(X ≤ x) by Sklar's theorem, where F is the marginal CDF. Both probabilities are regularly varying with index −ν as x → −∞ (Copula.Elliptical.StudentTTail.MixtureTail), so their ratio converges to the ratio of the Gaussian tail moments E[max(min(Z₁, Z₂), 0)^ν] / E[max(Z₁, 0)^ν], evaluated in polar coordinates (Copula.Elliptical.StudentTTail.Polar). The upper tail follows from radial symmetry.

Main results

• studentTTailCoeff ν r: the angular expression above.
• hasLowerTailDependence_studentT, hasUpperTailDependence_studentT.
• studentTTailCoeff_pos (r > −1), studentTTailCoeff_neg_one, studentTTailCoeff_one, studentTTailCoeff_le_one, studentTTailCoeff_eq_zero_iff.

References

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

noncomputable def ProbabilityTheory.Copula.studentTTailCoeff (ν r : ℝ) : ℝ

The tail-dependence coefficient of the bivariate Student-t copula, in angular form: λ(ν, r) = ∫_{arccos(r)/2}^{π/2} cos^ν θ dθ / ∫_0^{π/2} cos^ν θ dθ.

Equations

• ProbabilityTheory.Copula.studentTTailCoeff ν r = (∫ (θ : ℝ) in Real.arccos r / 2..Real.pi / 2, Real.cos θ ^ ν) / ∫ (θ : ℝ) in 0..Real.pi / 2, Real.cos θ ^ ν

theorem ProbabilityTheory.Copula.integral_cos_rpow_pos {ν a : ℝ} (hν : 0 < ν) (ha0 : 0 ≤ a) (ha : a < Real.pi / 2) : 0 < ∫ (θ : ℝ) in a..Real.pi / 2, Real.cos θ ^ ν

∫_a^{π/2} cos^ν > 0 for 0 ≤ a < π/2.

theorem ProbabilityTheory.Copula.normalTailMoment_pos {ν : ℝ} (hν : 0 < ν) : 0 < normalTailMoment ν

theorem ProbabilityTheory.Copula.tendsto_nhdsGT_zero_of_tendsto_cdfUnit {μ : MeasureTheory.Measure ℝ} [MeasureTheory.IsProbabilityMeasure μ] (hc : Continuous ↑(ProbabilityTheory.cdf μ)) {R : ↑unitInterval → ℝ} {l : ℝ} (hpos : ∀ᶠ (x : ℝ) in Filter.atBot, 0 < ↑(ProbabilityTheory.cdf μ) x) (h : Filter.Tendsto (fun (x : ℝ) => R (cdfUnit μ x)) Filter.atBot (nhds l)) : Filter.Tendsto R (nhdsWithin 0 (Set.Ioi 0)) (nhds l)

Passing from a limit along marginal quantiles x → −∞ to the limit t → 0⁺.

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

Lower tail dependence of the Student-t copula (Embrechts–McNeil–Straumann 2002, Hult–Lindskog 2002): λ_L = ∫_{arccos(r)/2}^{π/2} cos^ν / ∫_0^{π/2} cos^ν.

theorem ProbabilityTheory.Copula.hasUpperTailDependence_studentT {r : ℝ} (hr : r ∈ Set.Icc (-1) 1) {ν : ℝ} (hν : 0 < ν) : (studentT (corrMatrix r) ⋯ ⋯ ν hν).HasUpperTailDependence (studentTTailCoeff ν r)

Upper tail dependence of the Student-t copula: λ_U = λ_L, by radial symmetry.

Properties of the coefficient

theorem ProbabilityTheory.Copula.studentTTailCoeff_pos {ν r : ℝ} (hν : 0 < ν) (hr : -1 < r) : 0 < studentTTailCoeff ν r

The Student-t copula is tail dependent for every r > −1: λ(ν, r) > 0.

theorem ProbabilityTheory.Copula.studentTTailCoeff_neg_one (ν : ℝ) : studentTTailCoeff ν (-1) = 0

For r = −1 (the countermonotonic case) there is no tail dependence.

theorem ProbabilityTheory.Copula.studentTTailCoeff_one {ν : ℝ} (hν : 0 < ν) : studentTTailCoeff ν 1 = 1

For r = 1 (the comonotonic case) the coefficient is 1.

theorem ProbabilityTheory.Copula.studentTTailCoeff_eq_zero_iff {ν r : ℝ} (hν : 0 < ν) (hr : r ∈ Set.Icc (-1) 1) : studentTTailCoeff ν r = 0 ↔ r = -1

The coefficient vanishes exactly in the countermonotonic case r = −1.

theorem ProbabilityTheory.Copula.studentTTailCoeff_le_one {ν r : ℝ} (hν : 0 < ν) : studentTTailCoeff ν r ≤ 1

The coefficient is at most 1.