Bootstrap-Based Inference for Cube Root Consistent Estimators ∗ Matias D. Cattaneo † Michael Jansson ‡ Kenichi Nagasawa § April 27, 2017 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. Our method restores consistency of the nonparametric bootstrap by altering the shape of the criterion function defining the estimator whose distribution we seek to approximate. This modification leads to a generic and easy-to-implement resampling method for inference that is conceptually distinct from other available distributional approximations based on some form of modified bootstrap. We offer simulation evidence showcasing the performance of our inference method in finite samples. An extension of our methodology to general M- estimation problems is also discussed. Keywords: cube root asymptotics, bootstrapping, maximum score estimation, isotonic density estimation. JEL : C12, C14, C21. arXiv:1704.08066v1 [math.ST] 26 Apr 2017 Abstract $^{∗}$We thank Andres Santos for very thoughtful discussions. Cattaneo gratefully acknowledges financial support from the National Science Foundation through grant SES-1459931, and Jansson gratefully acknowledges financial support from the National Science Foundation through grant SES-1459967 and the research support of CREATES (funded by the Danish National Research Foundation under grant no. DNRF78). $^{†}$Department of Economics and Department of Statistics, University of Michigan. $^{‡}$Department of Economics, University of California at Berkeley and CREATES. $^{§}$Department of Economics, University of Michigan.