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arXiv · 2609.23313

Full negative regression dependence: counterexamples for tail dependence and negative association

Yuting Su·Taizhong Hu
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Abstract

We show that full negative regression dependence implies neither negative right-tail dependence nor negative left-tail dependence. A strictly positive distribution on a 2*2*3 rectangular state space provides an exact counterexample for the right-tail case, and its coordinatewise reflection yields a counterexample for the left-tail case. We further prove that this state-space size is minimal. The counterexample is established via a finite upper-set criterion using exact integer arithmetic. We also resolve the open question of whether full negative regression dependence implies negative association. A strictly positive distribution on {0,1,2}^4 satisfies full negative regression dependence yet violates negative association; this counterexample is verified by 17532 exact integer inequalities. The main results of this paper were first uncovered by the multi-agent mathematical system Eureka and later verified by the authors.

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Yuting Su, Taizhong Hu. 2026-09-20. Full negative regression dependence: counterexamples for tail dependence and negative association. https://arxiv.org/abs/2609.23313

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