We study causal effect estimation with compositional treatments, where the exposure lies on a simplex and the estimand is defined over compositions rather than scalar or binary values. By considering a projection of the average potential outcome onto the treatment space, a kernel-based covariate functional balancing approach is adopted for weight construction. The weights are obtained by directly minimizing a worst-case balancing error over a reproducing kernel Hilbert space (RKHS) defined on the joint space of treatments and covariates, instead of being estimated under a treatment assignment model. Building on these weights, an augmented weighted estimator (AWE) is proposed, where the outcome function is estimated via kernel ridge regression and combined with a marginal augmentation over the covariate distribution. Despite the complex structure of the resulting objective, a finite-dimensional convex optimization problem is formulated via a representer theorem and a low-rank approximation. The proposed estimator achieves $\sqrt{n}$-consistency without requiring consistent estimation or smoothness of the weights. An asymptotic normality result is established around a sample-specific target. Empirical performance is demonstrated through simulation studies and a real data application.

翻译：我们研究了成分化处理下的因果效应估计，其中暴露变量位于单纯形上，目标估计量针对成分组合而非标量或二元值定义。通过将平均潜在结果投影到处理空间，采用基于核的协变量函数平衡方法进行权重构造。该权重通过直接最小化在处理与协变量联合空间上定义的再生核希尔伯特空间（RKHS）中的最坏情况平衡误差获得，而非在处理分配模型下进行估计。基于这些权重，我们提出了一种增广加权估计量（AWE），其中结局函数通过核岭回归估计，并结合对协变量分布的边缘增广技术。尽管目标函数结构复杂，但通过表示定理和低秩近似可将其转化为有限维凸优化问题。所提出的估计量在不要求权重一致估计或光滑性的条件下达到$\sqrt{n}$一致性，并建立了围绕样本特异性目标估计量的渐近正态性结果。通过仿真研究和真实数据应用验证了其实证性能。