协变量非随机缺失回归的有限样本Hausdorff界与Hadamard敏感性
Finite-Sample Hausdorff Bounds and Hadamard Sensitivity for Regressions with MNAR Covariates
- CentraleSupélec（中央苏佩莱克高等学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对协变量非随机缺失的线性回归，通过紧致区间限制和Aumann期望刻画识别集，推导有限样本Hausdorff界，并引入Hadamard设计方法探索插补影响，实现根号n收敛。
AI中文摘要：
协变量非随机缺失通常在没有不可检验的限制条件下会妨碍回归系数的点识别。本文研究当每个缺失协变量被限制在预先指定的紧致区间内时的线性回归问题。由此产生的总体目标是与观测数据律相容的最佳线性预测系数集合。在非原子观测数据律和所述正则性条件下，我们可以将该识别集表示为最小二乘矩映射下随机矩集的Aumann期望的像。对于总体集及其经验类比，我们在有界、次指数和多项式包络条件下推导出显式界。这些界显示了它们对样本量、维度、置信水平和尾部参数的依赖性。在Donsker条件和一致零集误差界下，集值Z估计量的Oracle扩大也以根号n速率收敛。我们单独研究了一种Hadamard设计方法，用于探索可容许插补的影响而无需枚举所有顶点，并建立直觉以检测总体目标可被zonotope近似的情形。数值实验说明了共缺失性和插补区间宽度如何影响该诊断。
英文摘要：
Covariates missing not at random generally prevent point identification of regression coefficients without untestable restrictions. This paper studies linear regression when every missing covariate is restricted to a prespecified compact interval. The resulting population target is the set of best linear predictor coefficients compatible with the observed-data law. Under a non-atomic observed-data law and the stated regularity conditions, we can represent this identified set as the image, under the least-squares moment map, of the Aumann expectation of a random moment set. For the population set and its empirical analogue, we derive explicit bounds under bounded, sub-exponential and polynomial envelope conditions. The bounds show their dependence on sample size, dimension, confidence level and tail parameters. Under a Donsker condition and a uniform zero-set error bound, an oracle enlargement of the set-valued Z-estimator also converges at the root n rate. We separately study a Hadamard-design method for exploring the effect of admissible imputations without enumerating all vertices and build intuition to detect situations where the population target can be approximated by a zonotope. A numerical experiment illustrates how co-missingness and the width of the imputation intervals affect this diagnostic.
