We consider estimation and inference on average treatment effects under unconfoundedness conditional on the realizations of the treatment variable and covariates. Given nonparametric smoothness and/or shape restrictions on the conditional mean of the outcome variable, we derive estimators and confidence intervals (CIs) that are optimal in finite samples when the regression errors are normal with known variance. In contrast to conventional CIs, our CIs use a larger critical value that explicitly takes into account the potential bias of the estimator. When the error distribution is unknown, feasible versions of our CIs are valid asymptotically, even when $\sqrt{n}$-inference is not possible due to lack of overlap, or low smoothness of the conditional mean. We also derive the minimum smoothness conditions on the conditional mean that are necessary for $\sqrt{n}$-inference. When the conditional mean is restricted to be Lipschitz with a large enough bound on the Lipschitz constant, the optimal estimator reduces to a matching estimator with the number of matches set to one. We illustrate our methods in an application to the National Supported Work Demonstration.

翻译：我们考虑在没有根据的情况下对平均治疗效果的估算和推断,条件是实现治疗变量和共变值。鉴于对结果变量的有条件平均值的不对称平稳和/或形状限制,当回归误差正常且已知差异为正常时,我们在有限样本中得出最优的估测和信任间隔(CIs),与常规的CIs相比,我们的CIs使用一个更大的关键值,明确考虑到估计值的潜在偏差。当误差分布不明时,我们的CIs的可行版本是无效的,即使由于缺少重叠,或条件平均值的平滑度较低,也不可能做到对准。我们还根据对美元和已知差异所必需的有条件平均值,得出最低顺差条件。当条件平均值被限制为利普施奇茨,且对利普施奇茨常数的束缚足够大时,最佳估计值将降低为与设定的匹配数的估测值,即使由于缺少重叠,或条件平均值较低,也不可能实现。我们用“支持国家工作”演示应用的方法。