Adaptation to unknown regression-function smoothness

Develop a data-driven estimator of the constant conditional variance sigma^2 in homoskedastic nonparametric random-design regression that adapts to an unknown degree of smoothness of the regression function.

Background

The paper's rate guarantees assume that the regression smoothness parameter beta_b is known, since the proposed two-scale estimators use tuning parameters and bandwidths depending on beta_b. The authors explicitly identify data-driven adaptation to unknown smoothness as a separate unresolved problem.

References

In particular, our work assumes that the degree of smoothness of the regression function is known. Data-driven adaptation to unknown smoothness and inference for the variance estimate remain separate problems.

— Improved Variance Estimation in Homoskedastic Nonparametric Random-Design Regression via a Two-Scale Approach  (2609.08783 - Dobriban et al., 8 Sep 2026) in Discussion section