Infill asymptotics for a stochastic process model with measurement error

In spatial modeling the presence of measurement error, or "nugget", can have a big impact on the sample behavior of the parameter estimates. This article investigates the nugget effect on maximum likelihood estimators for a one-dimensional spatial model: Ornstein-Uhlenbeck plus additive white noise. Consistency and asymptotic distributions are obtained under infill asymptotics, in which a compact interval is sampled over a finer and finer mesh as the sample size increases. Spatial infill asymptotics have a very different character than the increasing domain asymptotics familiar from time series analysis. A striking effect of measurement error is that MLE for the Ornstein-Uhlenbeck component of the parameter vector is only fourth-root-n consistent, whereas the MLE for the measurement error variance has the usual root-n rate.