标题： 具有空间混杂的高斯随机场之间回归斜率的一致内填充估计能力 显示英文标题

标题： Consistent Infill Estimability of the Regression Slope Between Gaussian Random Fields Under Spatial Confounding

摘要： 在由一个未观测的空间过程引起的混杂下，估计两个空间过程回归中的斜率参数的问题在近期的统计文献中引起了广泛关注。然而，一个基本问题仍未得到解决：在空间混杂的情况下，何时该斜率可以一致地估计？现有见解大多基于经验或特定于估计量。本文中，我们刻画了高斯随机场（GRF）之间回归斜率一致可估性的条件。在固定域（内填）渐近性下，我们给出了使用协方差主不规则项比率这一新颖刻画的一致可估性充分条件，这些比率决定了暴露过程和混杂过程的相对局部行为。当可估性成立时，我们通过局部差分（对适当阶数的过程取离散差分或拉普拉斯算子）提供了一致估计量。利用关于Paley-Wiener空间的功能分析结果，我们进一步提供了易于验证的必要条件，用于判断斜率一致可估性与混杂因子和暴露因子相对谱尾部衰减之间的关系。作为副产品，我们建立了多元GRF路径上测度等价的一个新颖且通用的谱条件，这是一项具有独立重要性的结果。我们证明了对于Matérn族、幂指数族、广义Cauchy族和协同区域化族，必要的和充分的条件变得相同，从而为在空间混杂下斜率一致可估性提供了完整刻画。通过局部平均和差分估计量，我们将结果扩展到包含测量误差的情形。通过数值实验探索了有限样本行为。 显示英文摘要

摘要： The problem of estimating the slope parameter in regression between two spatial processes under confounding by an unmeasured spatial process has received widespread attention in the recent statistical literature. Yet, a fundamental question remains unsolved: when is this slope consistently estimable under spatial confounding, with existing insights being largely empirical or estimator-specific. In this manuscript, we characterize conditions for consistent estimability of the regression slope between Gaussian random fields (GRFs). Under fixed-domain (infill) asymptotics, we give sufficient conditions for consistent estimability using a novel characterization of the regression slope as the ratio of principal irregular terms of covariances, dictating the relative local behavior of the exposure and confounder processes. When estimability holds, we provide consistent estimators of the slope using local differencing (taking discrete differences or Laplacians of the processes of suitable order). Using functional analysis results on Paley-Wiener spaces, we then provide an easy-to-verify necessary condition for consistent estimability of the slope in terms of the relative spectral tail decays of the confounder and exposure. As a by-product, we establish a novel and general spectral condition on the equivalence of measures on the paths of multivariate GRFs with component fields of varying smoothnesses, a result of independent importance. We show that for the Mat\'ern, power-exponential, generalized Cauchy, and coregionalization families, the necessary and sufficient conditions become identical, thereby providing a complete characterization of consistent estimability of the slope under spatial confounding. The results are extended to accommodate measurement error using local-averaging-and-differencing based estimators. Finite sample behavior is explored via numerical experiments.