The density weighted average derivative (DWAD) of a regression function is a canonical parameter of interest in economics. Classical first-order large sample distribution theory for kernel-based DWAD estimators relies on tuning parameter restrictions and model assumptions that imply an asymptotic linear representation of the point estimator. These conditions can be restrictive, and the resulting distributional approximation may not be representative of the actual sampling distribution of the statistic of interest. In particular, the approximation is not robust to bandwidth choice. Small bandwidth asymptotics offers an alternative, more general distributional approximation for kernel-based DWAD estimators that allows for, but does not require, asymptotic linearity. The resulting inference procedures based on small bandwidth asymptotics were found to exhibit superior finite sample performance in simulations, but no formal theory justifying that empirical success is available in the literature. Employing Edgeworth expansions, this paper shows that small bandwidth asymptotic approximations lead to inference procedures with higher-order distributional properties that are demonstrably superior to those of procedures based on asymptotic linear approximations.

翻译：回归函数的密度加权平均导数（DWAD）是经济学中重要的规范参数。基于核的DWAD估计量的经典一阶大样本分布理论依赖于调整参数约束和模型假设，这些条件意味着点估计量的渐近线性表示。这些约束可能具有限制性，且由此得到的分布逼近可能无法反映目标统计量的实际抽样分布，尤其该逼近对带宽选择缺乏稳健性。小带宽渐近为基于核的DWAD估计量提供了更普适的替代分布逼近方法，该方法允许但不要求渐近线性性质。仿真研究表明，基于小带宽渐近的推断程序在有限样本中表现更优，但现有文献中尚无严谨理论证明这一实证成功。本文通过运用Edgeworth展开证明，小带宽渐近逼近所生成的推断程序具有高阶分布性质，其性能显著优于基于渐近线性逼近的方法。