Spearman's rho dominates Kendall's tau under positive tail monotonicity #

Capéraà and Genest (1993), Spearman's ρ is larger than Kendall's τ for positively dependent random variables, J. Nonparametric Statistics 2, 183–194; see also Nelsen, An Introduction to Copulas, 2nd ed., §5.2.3: if V is left tail decreasing and right tail increasing in U, then

0 ≤ τ ≤ ρ.

In particular this holds for stochastically increasing copulas, for LCSD ∧ RCSI copulas and for copulas with an MTP2 density (via Dependence.HierarchyDensity). Combined with the PQD bound ρ ≤ 3τ this gives τ ≤ ρ ≤ 3τ.

Proof #

Write D(u,v) = C(u,v) - uv. LTD says D(u,v)/u decreases in u, and RTI says D(u,v)/(1-u) increases in u. Hence, for every fixed v and u₀ ∈ (0,1), the section D(·,v) dominates the tent D(u₀,v) · min(u/u₀, (1-u)/(1-u₀)), whose integral is D(u₀,v)/2. So every value of D(·,v) is at most twice its mean:

D(u₀,v) ≤ 2 ∫₀¹ D(u,v) du.

Integrating this pointwise bound with respect to C (whose second marginal is uniform) gives ∫ D dC ≤ 2 ∫∫ D, which is exactly τ ≤ ρ since ρ - τ = 8 ∫∫ D - 4 ∫ D dC.

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theorem ProbabilityTheory.Copula.IsLTD.cdf_sub_mul_le_two_mul_integral {C : Copula 2} (hL : C.IsLTD) (hR : C.IsRTI) (u₀ v : ↑unitInterval) :

C.cdf ![u₀, v] - ↑u₀ * ↑v ≤ 2 * ∫ (u : ↑unitInterval), C.cdf ![u, v] - ↑u * ↑v

Under LTD and RTI every value of a section of C - Π is at most twice its mean.

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theorem ProbabilityTheory.Copula.IsLTD.kendallTau_le_spearmanRho {C : Copula 2} (hL : C.IsLTD) (hR : C.IsRTI) :

C.kendallTau ≤ C.spearmanRho

Capéraà–Genest. If V is left tail decreasing and right tail increasing in U, then Kendall's tau is at most Spearman's rho (Capéraà–Genest 1993; Nelsen, §5.2.3).

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