Goodness-of-fit tests for parametric circular regression with spatial dependence: an application to wave direction modeling

Abstract This paper introduces goodness-of-fit tests for parametric circular regression models with spatially correlated errors, motivated by the analysis of wave direction data in the Adriatic Sea. The proposed methodology confronts a parametric model with a local linear kernel estimator that accounts for both the circular nature of the response and the spatial dependence in the observed field. Two test statistics are introduced: one based on the raw discrepancy between the parametric and nonparametric fits, and another that incorporates a smoothed version of the parametric model to improve robustness. Under suitable regularity conditions, we derive the asymptotic distribution of the test statistic based on the smoothed parametric fit, providing theoretical support for its validity. Calibration under the null hypothesis is achieved via spatial bootstrap techniques specifically adapted to circular data. A comprehensive simulation study demonstrates that the proposed tests are well-calibrated across various dependence scenarios and exhibit strong power against a wide range of alternatives. Finally, the methodology is illustrated through a detailed application to Adriatic Sea wave direction data, where a physically motivated parametric model is evaluated using the proposed tests. The results reveal no evidence against the model, supporting its adequacy as a parsimonious representation of the underlying spatial structure.