Series ridge regression for spatial data on ℝ^d

This paper develops a general asymptotic theory of series ridge estimators for spatial data observed at irregularly spaced locations in a sampling region R_n ⊂ℝ^d. We adopt a stochastic sampling design that can generate irregularly spaced sampling sites in a flexible manner including both pure increasing and mixed increasing domain frameworks. Specifically, we consider a spatial trend regression model and a nonparametric regression model with spatially dependent covariates. For these models, we investigate the L^2-penalized series estimation of the trend and regression functions and establish (i) uniform and L^2 convergence rates and (ii) multivariate central limit theorems for general series estimators, (iii) optimal uniform and L^2 convergence rates for spline and wavelet series estimators, and (iv) show that our dependence structure conditions on the underlying spatial processes cover a wide class of random fields including Lévy-driven continuous autoregressive and moving average random fields.