Bergsma--Dassios sign covariance with ties: a nonnegative decomposition and sharp bounds
Abstract
We construct tie-symmetrised extensions D̃ and R̃ of the unnormalised Hoeffding and Blum--Kiefer--Rosenblatt functionals D and R and prove, for every real-valued bivariate law, the exact nonnegative decomposition $ τ^{*}(X,Y)=12\widetilde D(X,Y)+24\widetilde R(X,Y), \qquad \widetilde D,\widetilde R\geq0, \qquad \widetilde R(X,Y)=0\quad\Longleftrightarrow\quad X\perp\!\!\!\perp Y. $ For atomless margins the components reduce to D and R, whereas in the presence of ties the corresponding classical identity τ*=12D+24R can fail. The components arise from the Hoeffding projections of a symmetrised pair kernel. For the four strict/non-strict boundary versions Dε,η and Rε,η, we establish the sharp comparisons $ 9\widetilde D\geq\sum_{ε,η}D_{ε,η}, \qquad 9\widetilde R\geq\sum_{ε,η}R_{ε,η}. $ Consequently, τ*≥0 for every bivariate law, and the Bergsma--Dassios conjecture is settled: τ*=0 if and only if X⊥ ⊥ Y. The boundary comparisons also yield the universal bound τ*≥(8/3)R. Finite-table arguments, including an exact sum-of-squares certificate, establish the boundary inequalities, and weak-order quantisation transfers them to arbitrary laws. The factor $9$ in each comparison and the coefficient $24$ in τ*≥24R̃ are sharp. The usual permutation test is consistent against every fixed dependent alternative, without assumptions on the margins.
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BibTeX
@misc{bergsma2026bergsmadassios,
title = {{Bergsma--Dassios sign covariance with ties: a nonnegative decomposition and sharp bounds}},
author = {Wicher Bergsma and Angelos Dassios},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06529},
eprint = {2609.06529},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06529},
}
RIS
TY - UNPB
TI - Bergsma--Dassios sign covariance with ties: a nonnegative decomposition and sharp bounds
AU - Wicher Bergsma
AU - Angelos Dassios
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.06529
UR - https://arxiv.org/abs/2609.06529
AB - We construct tie-symmetrised extensions D̃ and R̃ of the unnormalised Hoeffding and Blum--Kiefer--Rosenblatt functionals D and R and prove, for every real-valued bivariate law, the exact nonnegative decomposition $ τ^{*}(X,Y)=12\widetilde D(X,Y)+24\widetilde R(X,Y), \qquad \widetilde D,\widetilde R\geq0, \qquad \widetilde R(X,Y)=0\quad\Longleftrightarrow\quad X\perp\!\!\!\perp Y. $ For atomless margins the components reduce to D and R, whereas in the presence of ties the corresponding classical identity τ^*=12D+24R can fail. The components arise from the Hoeffding projections of a symmetrised pair kernel. For the four strict/non-strict boundary versions D_ε,η and R_ε,η, we establish the sharp comparisons $ 9\widetilde D\geq\sum_{ε,η}D_{ε,η}, \qquad 9\widetilde R\geq\sum_{ε,η}R_{ε,η}. $ Consequently, τ^*≥0 for every bivariate law, and the Bergsma--Dassios conjecture is settled: τ^*=0 if and only if X⊥ ⊥ Y. The boundary comparisons also yield the universal bound τ^*≥(8/3)R. Finite-table arguments, including an exact sum-of-squares certificate, establish the boundary inequalities, and weak-order quantisation transfers them to arbitrary laws. The factor $9$ in each comparison and the coefficient $24$ in τ^*≥24R̃ are sharp. The usual permutation test is consistent against every fixed dependent alternative, without assumptions on the margins.
ER -