A Bootstrap Goodness-of-fit Test for the Variogram

Abstract

[Abstract]: This paper introduces a nonparametric bootstrap test for assessing the adequacy of parametric variogram models in geostatistical processes with a non-constant trend. Several goodness-of-fit procedures for variogram models have been developed under stationarity. In applied settings, however, the trend is often not constant and must be estimated prior to variogram analysis. A common practice is therefore to detrend the data and apply these procedures to the resulting residuals. This strategy induces systematic bias in the empirical variogram and compromises the validity of inference, particularly when the trend is estimated nonparametrically. To address this issue, we propose a bootstrap testing procedure that explicitly accounts for trend estimation by incorporating a bias-corrected variogram estimator obtained through an iterative local linear approach. The method accommodates both simple and composite null hypotheses for general parametric variogram models, with spatial independence arising as a special case. Unlike residual-based tests for independence or classical procedures for assessing specific parametric forms, the proposed approach corrects the bias induced by detrending and yields more reliable inference. Simulation studies show that the method achieves accurate calibration and substantially improved power across a wide range of dependence structures and trend configurations. Two real-data examples further illustrate its practical relevance when both the trend and the spatial dependence must be estimated from irregularly spaced observations. The methodology is implemented using readily available tools in the R environment, facilitating its use in applied spatial analysis.