  We consider a stochastic process model with time trend and measurement error.
We establish consistency and derive the limiting distributions of the maximum
likelihood (ML) estimators of the covariance function parameters under a
general asymptotic framework, including both the fixed domain and the
increasing domain frameworks, even when the time trend model is misspecified or
its complexity increases with the sample size. In particular, the convergence
rates of the ML estimators are thoroughly characterized in terms of the growing
rate of the domain and the degree of model misspecification/complexity.
