Adaptive selection of unknown drift smoothness

Develop a fully adaptive version of the minimax-transport framework that selects or accommodates the unknown Hölder smoothness level of the nonparametric posterior-drift functions, for example through model selection, cross-validation, or Lepski-type tuning across multiple spline sieve levels.

Background

The nonparametric extension assumes that the posterior-drift functions belong to a Hölder class with smoothness level α. The spline-sieve estimator and its risk bound require the sieve dimension to be chosen as a function of this α, so the procedure is not fully operational when the smoothness is unknown.

The authors note that adaptive procedures based on model selection, cross-validation, or Lepski-type tuning could address this issue, but they do not develop or analyze such a procedure. A solution would make the minimax transport framework applicable without prespecifying the regularity of the drift functions and would ideally establish corresponding finite-sample or asymptotic risk guarantees.

References

The estimator above relies on a user-specified smoothness level \alpha. In real applications, this smoothness is often unknown, so adaptive procedures (e.g., model-selection/cross-validation \citep{Birge1997modelselectionadaptive, BarronBirgeMassart1999, DingTarokhYang2018} or Lepski-type tuning \citep{Lepskii1992Asymptoticallyminimax,Birge2001Lepskimethod} across multiple sieve levels) may be preferable. Developing a fully adaptive version in the present minimax-transport framework is an interesting direction for future work.

— Transporting Trial Evidence Under Posterior Drift and Possible Hidden Confounding  (2608.17999 - Mao et al., 18 Aug 2026) in Remark following Theorem 3 (Section 4, subsection “Nonparametric drifts”)