Diameter of universal-inference confidence sets in incomplete models

Determine whether the diameter of the cross-fit universal-inference confidence set CS_n, constructed via split-sample likelihood-ratio testing for incomplete discrete choice models, attains the O_p(sqrt(log(1/α)/n)) rate known in regular models with point-identified parameters, or otherwise characterize its (excess) diameter in the partially identified setting.

Background

The paper proposes a split-sample and cross-fit likelihood-ratio procedure for universal inference in incomplete discrete choice models and defines CS_n by test inversion. The method guarantees finite-sample validity without complex regularity conditions.

In regular, point-identified models, Wasserman, Ramdas, and Balakrishnan (2020, Theorem 5) show that universal-inference confidence sets have diameter O_p(sqrt(log(1/α)/n)), matching optimal rates. However, parameters in incomplete models are typically only partially identified, raising uncertainty about whether analogous diameter results hold. The authors explicitly leave analyzing the (excess) diameter of CS_n in incomplete models for future work.

References

In incomplete models, θ is typically partially identified. As such, it is not immediately clear whether such a result can be extended to the current setting. To keep a tight focus, we leave the analysis of (excess) diameters of CS_n for future work.

— Universal Inference for Incomplete Discrete Choice Models  (2501.17973 - Kaido et al., 29 Jan 2025) in Remark following Corollary 3.1, Section 3 (Finite Sample Validity)

Under the specified outer confidence model, these constructions therefore show no observed sample-level sign-detection advantage. The common feasible set controls the information supplied to both procedures; separating confidence-region conservatism from intrinsic sampling difficulty remains an open statistical question.

— The Missing Corner: When Sharp Causal Bounds Fail to Compose  (2610.08197 - Qi et al., 6 Oct 2026) in Section 4, immediately following Table 1 (Section "Population sign loss and sampling")