Determining significant covariates is a fundamental problem in spatial regression analysis. However, parametric assumptions limit flexibility and can lead to inaccurate inference when misspecified. To address this, we propose a fully nonparametric testing procedure for spatial regression that does not impose restrictive model assumptions. Our approach follows a Monte Carlo testing framework through random shifts with both torus and variance correction. We construct test statistics based on the correlation between the residuals, where the effects of nuisance covariates have been removed, and the covariate of interest, allowing us to assess the significance of the covariate in the sense of partial correlation. This enables robust inference across various models as it does not require parametric assumptions about the distribution, spatial correlation structure, and form of the dependence or even a closed-form distribution of the test statistics. Our method is computationally stable compared to the parametric approaches that depend on numerical optimisation. Furthermore, we show that the random shift test with variance correction and sample covariance as the test statistic is asymptotically exact. Through extensive numerical experiments, we demonstrate that our method achieves the nominal significance level and exhibits power comparable to that of parametric methods, even when they are correctly specified.

翻译：确定显著协变量是空间回归分析中的基本问题。然而，参数化假设限制了灵活性，且在模型设定错误时可能导致不准确的推断。为解决这一问题，我们提出了一种完全非参数的空间回归检验程序，该程序无需施加严格的模型假设。我们的方法采用蒙特卡洛检验框架，通过带环面校正与方差校正的随机偏移来实现。我们构建了基于残差（已去除干扰协变量效应）与目标协变量之间相关性的检验统计量，从而能够以偏相关意义评估协变量的显著性。该方法无需对分布、空间相关结构、依赖形式乃至检验统计量的闭式分布做参数假设，因此可在各类模型中实现鲁棒推断。与依赖数值优化的参数方法相比，我们的方法计算稳定性更强。此外，我们证明具有方差校正的随机偏移检验以样本协方差作为检验统计量时具有渐近精确性。通过大量数值实验表明，即使在参数模型设定正确的情况下，我们的方法也能达到名义显著性水平，并展现出与参数方法相当的检验功效。