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Our results about the maximum score estimator can be summarized as follows. 

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 . 

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 

<img src='content_image/324996.jpg'>

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 

