Analytical characterization of the calibrated achievability bound

Determine whether the tightened calibrated achievability bound for composite binary hypothesis testing under a uniform Type I error constraint admits an analytical characterization and, consequently, a closed-form expression.

Background

The paper introduces a calibrated achievability procedure that fixes the log-likelihood ratio induced by a joint Rényi projection and numerically selects its threshold and boundary randomization to satisfy the composite Type I constraint exactly. This calibration can improve on the explicit exponential bound, but its optimization is less amenable to analysis because the threshold and projected pair may depend on the Rényi order.

The authors explicitly leave unresolved whether this tightened bound can be represented analytically or in closed form. This concerns the finite-sample evaluation of the bound rather than the already established asymptotic error exponents.

References

It remains to be seen whether the tightened bound admits an analytical characterisation and, consequently, a closed form expression.

— Finite Sample Bounds for Composite Hypothesis Testing  (2608.28068 - Vera-Sig{ü}enza et al., 28 Aug 2026) in Section VIII, Conclusion, p. 15

The proposed threshold may not be exactly minimax optimal for a fixed $\alpha$.

— Improved E-Value Thresholds with Applications to Multiple and Sequential Testing  (2610.11186 - Shen et al., 8 Oct 2026) in Section 3, subsection “Thresholds under Lipschitz Densities,” immediately before Theorem 3.2 (Small-α asymptotic sharpness)