The weighted average quantile derivative (AQD) is the expected value of the partial derivative of the conditional quantile function (CQF) weighted by a function of the covariates. We consider two weighting functions: a known function chosen by researchers and the density function of the covariates that is parallel to the average mean derivative in Powell, Stock, and Stoker (1989, Econometrica 57, 1403–1430). The AQD summarizes the marginal response of the covariates on the CQF and defines a nonparametric quantile regression coefficient. In semiparametric single-index and partially linear models, the AQD identifies the coefficients up to scale. In nonparametric nonseparable structural models, the AQD conveys an average structural effect under certain independence assumptions. Including a stochastic trimming function, the proposed two-step estimator is root-n-consistent for the AQD defined by the entire support of the covariates. To facilitate tractable asymptotic analysis, a key preliminary result is a new Bahadur-type linear representation of the generalized inverse kernel-based CQF estimator uniformly over the covariates in an expanding compact set and over the quantile levels. The weak convergence to Gaussian processes applies to the differentiable nonlinear functionals of the quantile processes.

加权平均分位数导数 (AQD) 是由协变量的函数加权的条件分位数函数 (CQF) 的偏导数的期望值。我们考虑两个加权函数：研究人员选择的已知函数和与 Powell、Stock 和 Stoker 中的平均平均导数平行的协变量密度函数 (1989, Econometrica57, 1403–1430)。AQD 总结了协变量对 CQF 的边际响应，并定义了一个非参数分位数回归系数。在半参数单指数和部分线性模型中，AQD 可按比例识别系数。在非参数不可分离结构模型中，AQD 在某些独立性假设下传达了平均结构效应。包括一个随机修剪函数，所提出的两步估计器对于由协变量的整个支持定义的 AQD是根n一致的。为了促进易于处理的渐近分析，一个关键的初步结果是广义逆核为基础的 CQF 估计量的新 Bahadur 型线性表示，其在扩展中的协变量上均匀紧集和超过分位数水平。高斯过程的弱收敛适用于分位数过程的可微非线性泛函。