Full negative regression dependence: counterexamples for tail dependence and negative association
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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BibTeX
@misc{su2026full,
title = {{Full negative regression dependence: counterexamples for tail dependence and negative association}},
author = {Yuting Su and Taizhong Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23313},
eprint = {2609.23313},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23313},
}
RIS
TY - UNPB
TI - Full negative regression dependence: counterexamples for tail dependence and negative association
AU - Yuting Su
AU - Taizhong Hu
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23313
UR - https://arxiv.org/abs/2609.23313
AB - 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.
ER -