Our results about the maximum score estimator can be summarized as follows. As in the case of Lemma 1, the proof of part (ii) of the theorem goes beyond consistency and develops a Nagar-type MSE expansion for ˜ V MS n under some the additional assumptions. The AMSE can be minimized by choosing h$_{n}$ proportional to n − 1 / $^{7}$; for details, including a characterization of the optimal factor of proportionality and a simple rule-of-thumb choice thereof based on a Gaussian reference model, see the supplemental appendix (Theorem A.3 and Section A.5). To investigate the finite sample properties of our proposed bootstrap-based inference procedures, we conducted a Monte Carlo experiment. Following Horowitz (2002), and to allow for a comparison with his bootstrap-based inference method for the smoothed maximum score estimator, we generate data from a model of the form (11) with d = 1 , where and where ε can take three distinct distributions. Specifically, DGP 1 sets ε ∼ Logistic (0 , 1) / √ 2 π $^{2}$/ 3 , DGP 2 sets ε ∼ T 3 $_{/}$√$_{3, where}$ T k denotes a Student’s t distribution with k degrees of freedom, and DGP 3 sets ε ∼ (1 + 2( x 1 + x 2 ) $^{2}$+ ( x 1 + x 2 ) $^{4}$) Logistic (0 , 1) / √ π $^{2}$/ 48 . The parameter is θ 0 = 1 in all cases. The Monte Carlo experiment employs a sample size n = 1 , 000 with B = 2 , 000 bootstrap replications and S = 2 , 000 simulations. For each of the three DGPs, we implement the stan- dard non-parametric bootstrap, m -out-of- n bootstrap, and our proposed method using the two estimators ˜ V MS n and ˜ V ND n of V 0 . We report empirical coverage for nominal 95% confidence inter- vals and their average interval length. For the case of our proposed procedures, we investigate their performance using both (i) a grid of fixed tuning parameter values (bandwidth/derivative step) around the MSE-optimal choice and (ii) infeasible and feasible AMSE-optimal choices of the x = ( x 1 , x 2 ) ′ ∼ N    0 1    ,    1 0 0 1    , x ⊥ ε, 10 Theorem 2 Suppose Condition MS holds. (i) Then Condition CRA holds for m = m $^{MS}$. Suppose also that ˆ θ MS n satisfies (8) and that Condition K holds. (ii) If h$_{n}$ → 0 and nh 3 n → ∞ , then ˜ V MS n → P V MS 0 .