Abstract: In this study, we consider the estimation of partially linear models for spatial data distributed over complex domains. We use bivariate splines over triangulations to represent the nonparametric component on an irregular two-dimensional domain. The proposed method is formulated as a constrained minimization problem that does not require constructing finite elements or locally supported basis functions. Thus, it allows an easier implementation of piecewise polynomial representations of various degrees and various smoothness over an arbitrary triangulation. Moreover, the constrained minimization problem is converted to an unconstrained minimization using a QR decomposition of the smoothness constraints, enabling us to develop a fast and efficient penalized least squares algorithm for fitting the model. The estimators of the parameters are proved to be asymptotically normal under some regularity conditions. The estimator of the bivariate function is consistent, and its rate of convergence is also established. The proposed method enables us to construct confidence intervals and permits inferences for the parameters. The performance of the estimators is evaluated using two simulation examples and a real-data analysis.

Key words and phrases: Bivariate splines, penalty, semiparametric regression, spatial data, triangulation.