This paper develops a regression framework for the direct estimation of integrated functionals of conditional outcome distributions. The proposed method, termed rectified linear unit (ReLU) regression, projects the ReLU-transformed outcome onto covariates and admits a closed-form estimator. Its population regression function coincides with the integrated conditional distribution function of the outcome, and its convex conjugate, obtained via the Legendre-Fenchel transformation, recovers the integrated conditional quantile function. Both the regression and its conjugate require only mild distributional assumptions and accommodate non-continuous outcomes. We establish the uniform asymptotic distribution of the estimator and develop inference for the conjugate functional via the delta method for Hadamard directionally differentiable maps. Building on these results, we establish identification and inference for average quantile treatment effects over arbitrary subintervals of probability levels. This broadens the set of distributional parameters available to empirical work.

翻译：本文建立了一个用于直接估计条件结果分布积分泛函的回归框架。所提出的方法称为ReLU回归，通过对协变量进行ReLU变换后的结果变量投影，并得到闭式估计量。其总体回归函数与条件结果分布积分函数一致，而通过勒让德-芬切尔变换获得的凸共轭函数则重构了条件分位数积分函数。该回归及其共轭函数仅需温和的分布假设，且能处理非连续结果变量。我们建立了估计量的渐近均匀分布，并通过哈达玛方向可微映射的delta方法发展了共轭泛函的推断方法。基于上述结果，我们建立了任意概率水平子区间上平均分位数处理效应的识别与推断框架，从而拓展了实证研究中可用的分布参数集合。