标题： 关于最小最大线性估计量的渐近性 显示英文标题

标题： On the Asymptotics of the Minimax Linear Estimator

摘要： 许多因果估计量，如在无混杂条件下的平均处理效应，可以表示为未知回归函数的连续线性泛函。 我们研究了一种加权估计量，该估计量通过最小最大过程设置权重：解决一个凸优化问题，该问题在最坏情况下的条件偏差与方差之间进行权衡。 尽管该方法日益广泛使用，但对该方法的一般根-$n$理论却有限。 本文填补了这一空白。 在常规条件下，我们证明最小最大线性估计量是根-$n$一致且渐近正态的，并推导了其渐近方差。 这些结果证明了在形成大样本置信区间时可以忽略最坏情况偏差，并使推断对函数类的缩放不那么敏感。 在适度的方差条件下，该估计量达到了半参数效率界限，因此文献中常用的增强步骤不是实现一阶最优性的必要步骤。 来自模拟和三个实证应用的证据，包括职业培训和最低工资政策，指向了一个简单的规则：在满足我们常规条件的设计中，标准误差置信区间就足够；否则，考虑偏差的区间仍然重要。 显示英文摘要

摘要： Many causal estimands, such as average treatment effects under unconfoundedness, can be written as continuous linear functionals of an unknown regression function. We study a weighting estimator that sets weights by a minimax procedure: solving a convex optimization problem that trades off worst-case conditional bias against variance. Despite its growing use, general root-$n$ theory for this method has been limited. This paper fills that gap. Under regularity conditions, we show that the minimax linear estimator is root-$n$ consistent and asymptotically normal, and we derive its asymptotic variance. These results justify ignoring worst-case bias when forming large-sample confidence intervals and make inference less sensitive to the scaling of the function class. With a mild variance condition, the estimator attains the semiparametric efficiency bound, so an augmentation step commonly used in the literature is not needed to achieve first-order optimality. Evidence from simulations and three empirical applications, including job-training and minimum-wage policies, points to a simple rule: in designs satisfying our regularity conditions, standard-error confidence intervals suffice; otherwise, bias-aware intervals remain important.