(Politis and Romano, 1994), whose applicability to cube root asymptotic problems was pointed out by Delgado, Rodriguez-Poo, and Wolf (2001). A related method is the rescaled bootstrap (D¨ umbgen, 1993), whose validity in cube root asymptotic M -estimation problems was established recently by Hong and Li (2017). In addition, case-specific smooth bootstrap methods have been proposed for leading examples such as maximum score estimation (Patra, Seijo, and Sen, 2015) and isotonic density estimation (Kosorok, 2008; Sen, Banerjee, and Woodroofe, 2010). Each of these methods can also be viewed as offering a “robust” alternative to the nonparametric bootstrap but, unlike our proposed approach, they all achieve consistency by modifying the distribution used to generate the bootstrap counterpart of the estimator whose distribution is being approximated. In contrast, our method achieves consistency by means of an analytic modification to the objective function used to construct the standard nonparametric bootstrap distributional approximation, and hence is conceptually distinct from the modifications employed in the existing literature. The note proceeds as follows. Section 2 is heuristic in nature and serves the purpose of outlining the main idea underlying our approach. The heuristics of Section 2 are then made rigorous in Section 3. Section 4 considers the maximum score estimator, illustrating the implications of our main results for that estimator and investigating the small sample properties of our proposed inference procedure in a simulation experiment. Two distinct extensions of our results, to M-estimators exhibiting an arbitrary rate of convergence and to the Grenander (1956) estimator of an isotonic density, respectively, are provided in Section 5. Simulation evidence for the latter example is also provided. All derivations and proofs have been collected in the supplemental appendix. Suppose θ 0 ∈ Θ ⊆ R d is an estimand admitting the characterization 2 Cube Root Asymptotics where z is a random vector of which a random sample { z$_{i}$ : 1 ≤ i ≤ n } is available, and where m is a known function. Studying estimation problem of this kind for non-smooth m , Kim and Pollard 2 θ 0 = argmax θ ∈ Θ M ( θ ) , M ( θ ) = E [ m ( z , θ )] ,