Robust asymptotic justification for triangular step-down products

Establish a joint robust asymptotic e-process result for the product of the q cluster-robust step-down regression e-processes associated with the lower-triangular subgroup statistic, including its validity under general clustered projections and its relationship to the triangular Gaussian Bayes factor.

Background

The lower-triangular Gaussian Bayes factor factorizes into q scalar e-processes for sequential regressions of each outcome on the preceding outcomes and the original regressors. This factorization suggests an analogous product of cluster-robust e-processes in the nonparametric clustered setting.

However, the paper’s asymptotic theorem establishes validity one regression at a time and does not establish a joint result for the product. The appendix explicitly identifies the robust asymptotic justification of this triangular statistic as unresolved.

References

Its robust asymptotic justification is a separate question. Although eq:LTproduct suggests the product of the $q$ cluster-robust $e$-processes $_G$ of the step-down regressions, Theorem~\ref{thm:asymptotic} covers one regression at a time and no joint result is proved here (compare Remark~\ref{rem:bonferroni}).

eq:LTproduct:

EGLT=∏k=1qEG(k),E_G^{\mathrm{LT}}=\prod_{k=1}^qE_G^{(k)},

— Anytime-Valid Inference in Linear Models with Clustered Outcomes: Exact Multivariate Tests and Cluster-Robust Confidence Sequences  (2610.25613 - Lindon, 19 Oct 2026) in Appendix F, Section F.1