Spatial smoothing, Nugget effect and infill asymptotics

For spatio-temporal regression models with observations taken regularly in time but irregularly over space, we investigate the effect of spatial smoothing on the reduction of variance in estimating both parametric and nonparametric regression functions. The processes concerned are stationary in time but may be nonstationary over space. Our study indicates that under the infill asymptotic framework, the existence of the so-called nugget effect in either regressor process or noise process is necessary for spatial smoothing to reduce the estimation variance. In particular the nugget effect in the regressor process may lead to a faster convergence rate in estimating nonparametric regression functions.