Exact finite-sample certification threshold with unknown variance

Determine the exact finite-sample certification threshold for the exogenous fixed-design Gaussian regression problem with unknown return variance, lying between the universal binary-KL necessary condition and the sufficient threshold achieved by the ordinary terminal t test.

Background

The paper studies certification of a signal against an economic boundary when the regression loading and innovation variance are unknown. For an exogenous fixed-design Gaussian regression, it derives a universal impossibility condition from binary relative entropy and a constructive sufficient condition based on the noncentral-t power of the ordinary terminal t test. These bounds do not coincide at finite sample sizes: the information-equivalent t threshold is larger than the binary-KL converse, with convergence to the known-variance Gaussian threshold only as the degrees of freedom increase.

The gap between the necessary and sufficient thresholds is explicitly identified as an unresolved finite-sample region. Resolving it would require characterizing the optimal finite-sample certification procedure, or otherwise determining the exact minimax information requirement, under unknown variance and the stated error and power constraints.

References

This produces an explicit finite-sample unresolved region between a universal necessary threshold and a concrete sufficient threshold.

— Certified Alpha Capacity: Statistical Evidence, Economic Lifetime, and Arbitrage under Decay  (2610.01115 - Bonacorsi, 1 Oct 2026) in Section ‘Unknown scale and predictable regressors,’ subsection ‘Fixed-design benchmark with unknown variance’