去偏与同时推断：基于残差化Walsh-Hadamard得分的异质性因子效应修饰变量

中文摘要 AI 辅助

在包含K个二元处理组件的因子随机化实验中，我们研究哪些基线协变量会修饰组件主效应和交互效应。我们通过条件Walsh-Hadamard对比τ_S到标准化协变量的总体线性投影系数θ*_{jS}，定义效应修饰单元(j,S)，其中j∈[p]且∅≠S⊆[K]。利用基线稳健的直接得分，我们为所有p(2^K-1)个单元开发频率推断。首先，我们证明了一个具有全局敏感性的去偏中心极限定理：残差化基线通过一个精确条件均值零扰动进入交叉拟合得分。因此，影响函数余项仅需基线估计器的逐折加权L2一致性，无需规定的多项式速率或干扰物乘积速率条件。一个单独的显式惩罚尺度条件控制基线对得分Lasso的影响。其次，在由有界稀疏回归原始导出的明确陈述的均匀干扰物、基线、学生化和影响数组条件下，我们建立了整个网格上的高维高斯近似，并通过Walsh谱乘子自助法进行校准。这产生了同时置信带和Romano-Wolf逐步下降选择修饰单元，具有强家族误差控制，无需效应遗传假设。渐近分析中因子设计固定，而协变量维度可以增长；所有2^K-1个对比被联合覆盖。

英文摘要

In factorial randomized experiments with $K$ binary treatment components, we study which baseline covariates modify component main effects and interactions. We define effect-modifier cells $(j,S)$, $j\in[p]$ and $\emptyset\neq S\subseteq[K]$, through the coefficients $θ^\ast_{jS}$ of population linear projections of conditional Walsh--Hadamard contrasts $τ_{S}$ onto standardized covariates. Using baseline-robust direct scores, we develop frequentist inference for all $p(2^K-1)$ cells. First, we prove a debiased central limit theorem with global insensitivity: the residualization baseline enters the cross-fitted score through an exactly conditionally mean-zero perturbation. The influence-function remainder therefore requires only foldwise weighted $L_2$ consistency of the baseline estimator, with no prescribed polynomial rate or nuisance product-rate condition. A separate, explicit penalty-scale condition controls the baseline's effect on the score Lasso. Second, under explicitly stated uniform nuisance, baseline, studentization, and influence-array conditions derived from bounded sparse-regression primitives, we establish a high-dimensional Gaussian approximation over the full grid, calibrated by a Walsh-spectral multiplier bootstrap. This yields simultaneous confidence bands and Romano--Wolf step-down selection of modified cells with strong familywise error control, without effect-heredity assumptions. The factorial design is fixed in the asymptotic analysis, while the covariate dimension may grow; all $2^K-1$ contrasts are covered jointly.

发表机构

• Juntendo University（顺天堂大学）
• Kyoto University（京都大学）

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