tuning parameter. 

Table 1 presents the main results, which are consistent across all three simulation designs. First, as expected, the standard nonparametric bootstrap (labeled “Standard”) does not perform well, leading to confidence intervals with an average 64% empirical coverage rate. Second, the m -out-of- n bootstrap (labeled “m-out-of-n”) performs somewhat better for small subsamples, but leads to very large average interval length of the resulting confidence intervals. Our proposed methods, on the other hand, exhibit excellent finite sample performance in this Monte Carlo experiment. Results employing the example-specific plug-in estimator ˜ V MS n are presented under the label “Plug-in” while results employing the generic numerical derivative estimator ˜ V ND n are reported under the label “Num Deriv”. Empirical coverage appears stable across different values of the tuning parameters for each method, with better performance in the case of ˜ V MS n . We conjecture that n = 1 , 000 is too small for the numerical derivative estimator ˜ V ND n to lead to as good inference performance as ˜ V MS n (e.g., note that the MSE-optimal choice ϵ MSE is greater than 1). Nevertheless, empirical coverage of confidence intervals constructed using our proposed bootstrap-based method is close to 95% in all cases except when ˜ V ND n is used with either the infeasible asymptotice choice ϵ AMSE or its estimated counterpart ˆ ϵ AMSE , and with an average interval length of at most half that of any of the m -out-of- n competing confidence intervals. In particular, confidence intervals based on ˜ V MS n implemented with the feasible bandwidth ˆ h AMSE perform quite well across the three DGPs considered. 

In sum, applying the bootstrap-based inference methods proposed in this note to the case of the Maximum Score estimator of Manski (1975) lead to confidence intervals with very good coverage and length properties in the simulation designs considered. 

## 5 Extensions 

The scope of some of the main insights of this note extends beyond the 3 √$_{n}$$_{-consistent}$ M -estimators covered by Theorem 1. This section briefly discusses two possible extensions. First, we indicate how our main results can be generalized to M -estimators exhibiting an arbitrary rate of convergence. Second, we illustrate how the idea of reshaping can be used to achieve consistency of a bootstrap- based approximation to the distribution of another prominent 3 √$_{n}$$_{-consistent estimator, the isotonic}$ density estimator, which is not of M -estimator type. 

