Bandwidth selection methods for non parametric regression with spatially correlated data

In non parametric regression estimation, a critical and unavoidable step is to choose the smoothing parameters (bandwidth) to control the smoothness of the curve estimate. In the presence of spatially correlated errors, the traditional data-driven bandwidth selection methods, such as cross-validation and generalized cross-validation, do not work well for providing efficient bandwidth values. Moreover, the existing methods are based on estimation of correlation structure. As errors are unobservable, estimation of correlation is a big challenge. This article studies bandwidth selection methods for local linear regression (LLR) in the presence of correlated errors. For this purpose, we derive the weighted mean average square errors (WMASE) and use it along with bias-correlated generalized cross-validation (GCV) as bandwidth selection criterion. We then get the estimates of bandwidth by two ways: by minimizing the weighted mean average square errors (WMASE) and by using the bias-correlated generalized cross-validation (GCV) criterion. Due to its good properties, the composite likelihood (CL) is used to estimate the correlation. The results show that composite likelihood provides better estimates than maximum likelihood (ML) in the sense of producing more reasonable estimates of correlation parameters and bandwidth values. Also, our methods appear reasonably robust to the misspecification of the parametric correlation model. Moreover, the proposed methods work much better than ignoring the correlation and applying traditional methods. Finally, we apply our methods on water chemistry data collected about lakes in the Northeastern United States.