Abstract: In this study, we investigate varying-coefficient models for spatial data distributed over two-dimensional domains. First, we approximate the univariate components and the geographical component in the model using univariate polynomial splines and bivariate penalized splines over triangulation, respectively. The spline estimators of the univariate and bivariate functions are consistent, and their convergence rates are also established. Second, we propose empirical likelihood-based test procedures to conduct both pointwise and simultaneous inferences for the varying-coefficient functions. We derive the asymptotic distributions of the test statistics under the null and local alternative hypotheses. The proposed methods perform favorably in finite-sample applications, as we show in simulations and an application to adult obesity prevalence data in the United States.

Key words and phrases: B-spline, bivariate spline, empirical likelihood, Geo data, non-parametric hypothesis testing.