Higher-order efficient estimators extend standard first-order semiparametric estimators by replacing second-order residuals with third- or higher-order terms, potentially enabling asymptotic efficiency under slower nuisance function convergence rates and improving finite-sample performance. Existing methods achieve higher-order expansions through structurally different approximation strategies, including basis truncation, kernel smoothing, and highly adaptive lasso (HAL) representations, making direct theoretical and practical comparison difficult. In this manuscript, we provide a focused review and a simulation-based empirical benchmark for second-order efficient estimators, using treatment-specific mean estimation as a canonical causal inference and missing data problem. We compare how higher-order influence function (HOIF) estimators, kernel-based higher-order targeted minimum loss-based estimator (HOTMLE), and HAL-based HOTMLE construct higher-order expansions and the approximation or regularization burdens they introduce. The asymptotic and numerical study evaluates first-order and empirical second-order estimators under controlled nuisance errors with constant or increasing sectional variation complexity. Results show that higher-order debiasing can substantially reduce first-order estimation bias; however, gains depend strongly on stability of the approximation or regularization required for higher-order correction. Empirical HAL-based HOTMLE shows relatively stable performance, while empirical HOIF remains sensitive to basis truncation and tuning choices. Overall, this manuscript clarifies when higher-order asymptotic improvements are attained in theory, when they may be practically visible, and when implementation instability may offset theoretical advantages.

翻译：高阶高效估计量通过将二阶残差替换为三阶或更高阶项，扩展了标准一阶半参数估计量，从而在更慢的冗余函数收敛速度下实现渐近效率，并改善有限样本性能。现有方法通过结构不同的逼近策略实现高阶展开，包括基截断、核平滑和高度自适应Lasso（HAL）表示，这使得直接的理论与实践比较变得困难。本文以处理组特定均值估计作为因果推断与缺失数据问题的典型场景，对二阶高效估计量进行重点综述及基于模拟的经验基准研究。我们比较了高阶影响函数（HOIF）估计量、基于核的高阶定向最小损失估计量（HOTMLE）以及基于HAL的HOTMLE构建高阶展开的方式及其引入的逼近或正则化负担。在控制冗余误差（具有恒定或递增截面变化复杂度）的条件下，渐近与数值研究评估了一阶和经验二阶估计量。结果表明，高阶去偏可显著减少一阶估计偏差，但其增益高度依赖于高阶校正所需的逼近或正则化稳定性。基于HAL的经验HOTMLE表现出相对稳定的性能，而经验HOIF对基截断和调参选择仍较为敏感。总体而言，本文明确了高阶渐近改进在理论上何时可实现、实践中何时可能显现，以及实现中的不稳定性何时可能抵消理论优势。