A computational validation for nonparametric assessment of spatial trends

The analysis of continuously spatially varying processes usually considers two sources of variation, namely, the large-scale variation collected by the trend of the process, and the small-scale variation. Parametric trend models on latitude and longitude are easy to fit and to interpret. However, the use of parametric models for characterizing spatially varying processes may lead to misspecification problems if the model is not appropriate. Recently, Meilan-Vila et al. (TEST 29:728–749, 2020) proposed a goodness-of-fit test based on an $$L_2$$ -distance for assessing a parametric trend model with correlated errors, under random design, comparing parametric and nonparametric trend estimates. The present work aims to provide a detailed computational analysis of the behavior of this approach using different bootstrap algorithms for calibration, one of them including a procedure that corrects the bias introduced by the direct use of the residuals in the variogram estimation, under a fixed design geostatistical framework. Asymptotic results for the test are provided and an extensive simulation study, considering complexities that usually arise in geostatistics, is carried out to illustrate the performance of the proposal. Specifically, we analyze the impact of the sample size, the spatial dependence range and the nugget effect on the empirical calibration and power of the test.