Optimal Rates of Convergence of Parameter Estimators in the Binary Response Model with Weak Distributional Assumptions
证明在弱分布假设下，平滑最大得分估计量的收敛速度N^{-h/(2h+1)}是二元响应模型系数向量估计中可达的最快速度，从而确认了该估计量的最优性。
The smoothed maximum score estimator of the coefficient vector of a binary response model is consistent and, after centering and suitable normalization, asymptotically normally distributed under weak assumptions [5]. Its rate of convergence in probability is N − h /(2 h +1) , where h ≥ 2 is an integer whose value depends on the strength of certain smoothness assumptions. This rate of convergence is faster than that of the maximum score estimator of Manski [11,12], which converges at the rate N −1/3 under assumptions that are somewhat weaker than those of the smoothed estimator. In this paper I prove that under the assumptions of smoothed maximum score estimation, N − h /(2 h +1) is the fastest achievable rate of convergence of an estimator of the coefficient vector of a binary response model. Thus, the smoothed maximum score estimator has the fastest possible rate of convergence. The rate of convergence is defined in a minimax sense so as to exclude superefficient estimators.