Bootstrap validity for SLDB inference

Establish the validity of the standard nonparametric bootstrap for the sliced $L^p$ distributional balancing estimator and its associated average treatment effect estimator.

Background

The SLDB procedure uses empirical sorting and optimization over the simplex, making the smoothness and local perturbation properties required for standard bootstrap validity difficult to verify. The paper explicitly declines to establish bootstrap validity and reports simulation evidence that bootstrap confidence intervals can undercover. Consequently, the validity of bootstrap inference for SLDB remains unresolved, while Wald-type and subsampling procedures are recommended instead.

References

However, our SLDB estimation procedure involves empirical sorting and optimization over the simplex, making it nontrivial to formally establish the required regularity conditions. Therefore, we do not establish bootstrap validity for the SLDB estimator and urge caution when applying the bootstrap for any choice of $p$.

— Sliced $L^p$ Distributional Balancing  (2609.09600 - Zhang et al., 9 Sep 2026) in Section 5.2.3, Bootstrap