Determine optimal antisymmetric test statistics and compare them with the model-X GRO e-variable

Determine which importance statistics yield optimal antisymmetric e-variables for sequential conditional-independence testing and establish how those e-variables compare with the model-X growth-rate-optimal e-variable.

Background

The eCRT constructs antisymmetric e-values from an importance statistic, such as the predictive loss of a machine-learning model, under an exchangeability null. This approach is valid but tests a larger null than the original model-X conditional-independence null and may therefore discard information.

The paper discusses particular exponential and coin-betting choices and later derives optimality results for the exchangeability null. Nevertheless, the general question posed at the outset—especially which practical statistics are optimal and how antisymmetric procedures compare with the model-X GRO e-variable—was explicitly identified as unresolved.

References

Still, it is not clear which statistics are optimal, or how these antisymmetric e-values compare to $E{\textnormal{X-CI}$.

— Sequential Conditional Independence Testing with Machine Learning Models  (2610.11388 - Reyero-Lobo et al., 8 Oct 2026) in Section 2, Problem statement

In practice, the GRO e-variable would be estimated via kernel density estimation methods whose parameters such as the kernel family and bandwidth could be chosen to optimize e-power instead of density estimation, and thus possibly yield a worse density approximation but larger e-power for $B>1$. We leave such questions for future research.

— Sequential Conditional Independence Testing with Machine Learning Models  (2610.11388 - Reyero-Lobo et al., 8 Oct 2026) in Appendix, Section titled “Practical effects of derandomization”