A Most Powerful Test for the Adequateness of an Asymptotic Spatial Regression Model when the Observation is Disturbed by the Set-Indexed Brownian Sheet

Abstract: In this work we establish an optimal test for checking the appropriateness of a spatial regression model. In the study of model check for regression, the correctness of an assumed model is investigated by the partial sums of the residuals. In this work an inverted procedure is proposed in that we firstly embed the observation into a partial sums process to get the corresponding asymptotic regression model. Instead of considering the residuals of the model, we derive the Cameron-Martin density of the observation. For simple hypotheses under H0 as well as under H1 we derive the Neyman-Person test based on the ratio of the densities under H0 and H1. Interestingly, the rejection region can be exactly computed as an integral with respect to the partial sums process of the observation. An application of the procedure to a real data is also discussed.