Nonparametric covariance estimation with bias correction for spatial lattice processes

2026-10-08

AI Summary

This study addresses the substantial finite-sample bias inherent in nonparametric covariance estimation for spatial lattice processes, which severely compromises inferential accuracy under strong spatial dependence. We derive, for the first time, an exact analytical expression for the finite-sample bias of this estimator when the mean is unknown, and accordingly propose a joint bias-corrected estimator. The methodological validity is rigorously established through asymptotic normality theory and Monte Carlo simulations. The proposed approach significantly reduces both estimation bias and mean squared error in strongly dependent spatial settings, achieving approximately unbiased, high-precision covariance estimation. An empirical analysis using terrain data further demonstrates the practical utility of this method for real-world spatial statistical modeling.

Abstract

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.

Problem

• nonparametric covariance estimation
• bias correction
• spatial lattice processes
• finite-sample bias
• stationary processes

Innovation

• Nonparametric covariance estimation
• Bias correction
• Spatial lattice processes
• Finite-sample bias
• Asymptotic normality