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Kendall's tau of the Plackett family: integral representations

Kendall's tau of the Plackett copula C_θ (Copula.Families.Plackett.Basic; Nelsen 2006, §3.3.1) has no elementary closed form. This file proves two explicit integral representations.

• The conditional distribution functions of C_θ are the explicit partial derivatives: ∂C_θ/∂u (u, v) = plackettDeriv θ v u for almost every u (conditionalCDF_plackett), and by kendallTau_conditional_product and exchangeability (kendallTau_plackett_eq_integral)

  τ(C_θ) = 1 - 4 ∫₀¹∫₀¹ ∂_u C_θ(u,v) ∂_v C_θ(u,v) du dv,

  with ∂_v C_θ(u,v) = (1 - (1 + (θ-1)(u+v) - 2θu)/√disc(u,v))/2 (plackettDeriv).

• A rational representation (kendallTau_plackett_eq_rational, θ ≠ 1): writing disc(u,v) = [1 + (θ-1)(u+v)]² - 4θ(θ-1)uv (plackettDisc),

  τ(C_θ) = (θ+1)/(θ-1) - 2θ/(θ-1) ∫₀¹∫₀¹ (1 + (θ-1)(u+v-2uv)) / disc(u,v) du dv.

  It follows from the pointwise identity (four_mul_plackettDeriv_mul) 4 ∂_uC ∂_vC = -2/(θ-1) - 2(1-u-v)/√disc + 2θ(1 + (θ-1)(u+v-2uv))/((θ-1) disc) and the vanishing of ∫∫ (1-u-v)/√disc under the radial reflection (u,v) ↦ (1-u,1-v), which leaves disc invariant (integral_plackettOddPart).

A one-dimensional arctangent-integral form is in Copula.Families.Plackett.KendallArctan; strict monotonicity, sign and limits of τ(C_θ) are in Copula.Families.Plackett.KendallOrder.

Partial derivatives for all parameters

theorem ProbabilityTheory.Copula.plackettDeriv_one (u v : ℝ) : plackettDeriv 1 u v = u

For θ = 1 the derivative formula reduces to ∂(uv)/∂v = u.

theorem ProbabilityTheory.Copula.hasDerivAt_plackettCDF_right_of_mem {θ u v : ℝ} (hθ : 0 < θ) (hu0 : 0 ≤ u) (hu1 : u ≤ 1) (hv0 : 0 ≤ v) (hv1 : v ≤ 1) : HasDerivAt (fun (y : ℝ) => plackettCDF θ u y) (plackettDeriv θ u v) v

∂C_θ/∂v (u,v) = plackettDeriv θ u v on the closed unit square, for every θ > 0 (including θ = 1).

theorem ProbabilityTheory.Copula.measurable_plackettDeriv (θ : ℝ) : Measurable fun (q : ↑unitInterval × ↑unitInterval) => plackettDeriv θ ↑q.1 ↑q.2

theorem ProbabilityTheory.Copula.continuous_plackettDeriv {θ : ℝ} (hθ : 0 < θ) : Continuous fun (q : ↑unitInterval × ↑unitInterval) => plackettDeriv θ ↑q.1 ↑q.2

theorem ProbabilityTheory.Copula.conditionalCDF_plackett {θ : ℝ} (hθ : 0 < θ) (v : ↑unitInterval) : ∀ᵐ (u : ↑unitInterval), (plackett θ hθ).conditionalCDF u v = plackettDeriv θ ↑v ↑u

The conditional distribution function of the Plackett copula is the explicit partial derivative ∂C_θ/∂u (u, v) = plackettDeriv θ v u, for almost every u.

The partial-derivative representation

theorem ProbabilityTheory.Copula.kendallTau_plackett_eq_integral {θ : ℝ} (hθ : 0 < θ) : (plackett θ hθ).kendallTau = 1 - 4 * ∫ (v : ↑unitInterval) (u : ↑unitInterval), plackettDeriv θ ↑v ↑u * plackettDeriv θ ↑u ↑v

Kendall's tau of the Plackett copula as a double integral of the product of the explicit partial derivatives: τ(C_θ) = 1 - 4 ∫∫ ∂_uC_θ ∂_vC_θ.

The rational representation

theorem ProbabilityTheory.Copula.plackettDisc_reflect (θ u v : ℝ) : plackettDisc θ (1 - u) (1 - v) = plackettDisc θ u v

noncomputable def ProbabilityTheory.Copula.plackettOddPart (θ u v : ℝ) : ℝ

The odd part (1 - u - v)/√disc of the integrand.

Equations

• ProbabilityTheory.Copula.plackettOddPart θ u v = (1 - u - v) / √(ProbabilityTheory.Copula.plackettDisc θ u v)

noncomputable def ProbabilityTheory.Copula.plackettRatPart (θ u v : ℝ) : ℝ

The rational part (1 + (θ-1)(u+v-2uv))/disc of the integrand (the Plackett density times √disc / θ).

Equations

• ProbabilityTheory.Copula.plackettRatPart θ u v = (1 + (θ - 1) * (u + v - 2 * u * v)) / ProbabilityTheory.Copula.plackettDisc θ u v

theorem ProbabilityTheory.Copula.four_mul_plackettDeriv_mul {θ u v : ℝ} (hθ : θ ≠ 1) (hD : 0 < plackettDisc θ u v) : 4 * (plackettDeriv θ v u * plackettDeriv θ u v) = -2 / (θ - 1) - 2 * plackettOddPart θ u v + 2 * θ / (θ - 1) * plackettRatPart θ u v

The pointwise decomposition 4 ∂_uC ∂_vC = -2/(θ-1) - 2(1-u-v)/√disc + 2θ(1 + (θ-1)(u+v-2uv))/((θ-1) disc).

theorem ProbabilityTheory.Copula.continuous_plackettOddPart {θ : ℝ} (hθ : 0 < θ) : Continuous fun (q : ↑unitInterval × ↑unitInterval) => plackettOddPart θ ↑q.1 ↑q.2

theorem ProbabilityTheory.Copula.continuous_plackettRatPart {θ : ℝ} (hθ : 0 < θ) : Continuous fun (q : ↑unitInterval × ↑unitInterval) => plackettRatPart θ ↑q.1 ↑q.2

theorem ProbabilityTheory.Copula.integral_plackettOddPart (θ : ℝ) : ∫ (v : ↑unitInterval) (u : ↑unitInterval), plackettOddPart θ ↑u ↑v = 0

The odd part integrates to zero: the radial reflection (u,v) ↦ (1-u,1-v) preserves the uniform measure and disc, and changes the sign of 1 - u - v.

theorem ProbabilityTheory.Copula.kendallTau_plackett_eq_rational {θ : ℝ} (hθ : 0 < θ) (hθ1 : θ ≠ 1) : (plackett θ hθ).kendallTau = (θ + 1) / (θ - 1) - 2 * θ / (θ - 1) * ∫ (v : ↑unitInterval) (u : ↑unitInterval), (1 + (θ - 1) * (↑u + ↑v - 2 * ↑u * ↑v)) / plackettDisc θ ↑u ↑v

Rational integral representation of Kendall's tau of the Plackett copula (θ ≠ 1): τ(C_θ) = (θ+1)/(θ-1) - 2θ/(θ-1) ∫₀¹∫₀¹ (1 + (θ-1)(u+v-2uv))/disc(u,v) du dv.