The problem of assessing a parametric regression model in the presence of spatial correlation is addressed in this work. For that purpose, a goodness-of-fit test based on a $L_2$-distance comparing a parametric and a nonparametric regression estimators is proposed. Asymptotic properties of the test statistic, both under the null hypothesis and under local alternatives, are derived. Additionally, a bootstrap procedure is designed to calibrate the test in practice. Finite sample performance of the test is analyzed through a simulation study, and its applicability is illustrated using a real data example.

翻译：本研究探讨了在存在空间相关性的情况下评估参数回归模型的问题。为此，提出了一种基于$L_2$距离的拟合优度检验方法，通过比较参数回归估计量与非参数回归估计量来构建检验统计量。推导了该检验统计量在原假设和局部备择假设下的渐近性质。此外，设计了自助法（bootstrap）流程以在实际应用中对检验进行校准。通过模拟研究分析了检验在有限样本下的表现，并利用实际数据示例展示了其适用性。