# Bootstrap-Based Inference for Cube Root Consistent Estimators ∗ 

Matias D. Cattaneo † Michael Jansson ‡ Kenichi Nagasawa § 

April 27, 2017 

## Abstract 

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. 

