Before revision: This note proposes a consistent bootstrap-based distributional approximation for cube root consistent estimators such as the maximum score estimator of Manski (1975)and the isotonic density estimator of Grenander (1956). In both cases , the standard nonparametric bootstrap is known to be inconsistent.
After revision: This note proposes a consistent bootstrap-based distributional approximation for cube root consistent and related estimators exhibiting a Chernoff (1964)-type limiting distribution. For estimators of this kind , the standard nonparametric bootstrap is known to be inconsistent.