Un contre-exemple à une conjecture de Hutchinson et Lai

A sharp inequality for Kendall's $τ$ and Spearman's $ρ$ of Extreme-Value Copulas

“…x,y (T x 1 ,y 1 ), T x 1 ,y 1 ) ≥ 0 (11) holds. Moreoever, we have equality in (11) if and only if ϕ x 1 x,y (T x 1 ,y 1 ) is a triangular Pickands dependence function too, i.e. if ϕ x 1…”

“…, x n , x n+1 ) with x 0 := 0 < x 1 < x 2 < • • • < x n < x n+1 < 1 =: x n+2 , fix B ∈ A x and set y 1 = B(x 1 ). Since B can be represented as B = ϕ (x 2 ,...,x n+1 ) x 1 ,y 1 (A) for some piecewise linear A with at most n vertices, applying inequality (7) and inequality (11) yields The family of all piecewise linear Pickands functions is dense in (A, • ∞ ). The remaining assertion is a direct consequence of the identities τ (T x 1 ,y 1 ) = 1−y 1 y 1 and ρ(T x 1 ,y 1 ) = 3(1−y 1 ) 1+y 1 .…”

“…In the sequel we will refer to the first inequality in (1) as lower Hutchinson-Lai inequality and to the second one as upper Hutchinson-Lai inequality. A counterexample to the upper Hutchinson-Lai inequality (concerning the 3τ (X,Y ) 2 part) can be found in [12], and the lower Hutchinson-Lai inequality was disproved in [11]. On the other hand, [6] provided a variational calculus based proof of the Hutchinson-Lai inequalities for an important family of stochastically increasing (continuous) random variables (X, Y ) -those whose underlying copula C A is an Extreme-Value copula.…”