发表机构
TU Dortmund University（多特蒙德工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对$\boldsymbol{Z}^2$上平稳格点过程，推导样本协方差估计量的有限样本偏差公式，提出联合偏差校正的几乎无偏协方差估计量，经模拟和地形数据克里金应用验证了其降低偏差等优势。
AI 中文摘要
准确的协方差估计对空间数据分析至关重要。参数方法可能因模型误设导致错误结论，而非参数方法因依赖大量协方差参数估计且常面临有限样本偏差问题，在实践中常被忽视。本文研究当真实均值参数未知需估计时，$\boldsymbol{Z}^2$上平稳格点过程的样本协方差估计量的偏差性质。我们推导了样本协方差估计量的有限样本偏差的精确公式，证明其期望是由空间滞后和样本量决定的总体协方差的线性组合。基于此特征，我们提出联合偏差校正的协方差估计量，其几乎无偏。此外，我们推导了均方误差公式，并证明渐近正态性结果，表明带偏差校正和不带偏差校正的估计量具有渐近等价性。模拟结果显示偏差大幅降低，且在均方误差方面也常有所改善，尤其在强空间依赖情况下。对地形数据的克里金应用进一步说明了所提方法的实际益处。
英文摘要
Accurate covariance estimation is crucial for spatial data analysis. While parametric methods can suffer from model misspecification leading to wrong conclusions, nonparametric approaches are often neglected in practice as they rely on the estimation of a large number of covariance parameters and often face finite-sample bias issues. In this paper, we study the bias properties of sample covariance estimators for stationary lattice processes on $\mathbb{Z}^2$, when the true mean parameter is unknown and has to be estimated. We derive exact formulas for the finite-sample biases of sample covariance estimators and show that their expectations are linear combinations of population covariances determined by spatial lag and sample size. Based on this characterization, we propose jointly bias-corrected covariance estimators that are nearly unbiased. Additionally, we derive formulas for the mean-squared error and prove asymptotic normality results that show asymptotic equivalence for the estimators with and without bias correction. Simulations demonstrate substantial reductions of bias and often also in terms of MSE, particularly under strong spatial dependence. A kriging application to topography data further illustrates the practical benefits of the proposed approach.
