Asymptotic order of the covariance-adaptive BH bound

Determine whether, under suitable common-factor asymptotics for equi-correlated and common-factor Gaussian models, the covariance-adaptive finite-sample bound for the false discovery rate of the original Benjamini–Hochberg procedure recovers the optimal asymptotic order established by Lei (2026) while providing sharper finite-sample information over practically relevant correlation regimes.

Background

The paper derives covariance-specific lower and upper bounds for the finite-sample false discovery rate of the original Benjamini–Hochberg procedure in two-sided Gaussian mean testing. These bounds depend on the conditional variance parameters τ_i = 1 − R_i² and can be more informative than the distribution-free Benjamini–Yekutieli bound for specified covariance structures.

The authors contrast their finite-sample, covariance-specific bounds with asymptotic results for common-factor Gaussian models, including the optimal-order result of Lei (2026). They explicitly leave unresolved whether their covariance-adaptive bound has the same optimal asymptotic order under suitable common-factor asymptotic regimes while retaining sharper finite-sample behavior.

References

It is therefore of interest to determine whether, under suitable common-factor asymptotics, this covariance-specific bound recovers the optimal order identified by \citet{Lei2026}, while potentially providing sharper finite-sample information over practically relevant correlation regimes.

— Controlling the False Discovery Rate Control in Two-Sided Gaussian Mean Testing Under Arbitrary Dependence  (2608.21267 - Ghosh et al., 21 Aug 2026) in Remark following Theorem 4, Section 3, “Dependence-adaptive bounds for the BH FDR”; reiterated in Section 6, “Discussion”