An example of inconsistent MLE of spatial covariance parameters under increasing domain asymptotics

Asymptotic properties of estimators of covariance parameters in spatial statistics are commonly considered under the frameworks of increasing domain and fixed domain asymptotics, respectively. Although inconsistency is a general conclusion under the framework of fixed domain asymptotics, it is generally believed that consistency should generally hold under the framework of increasing domain asymptotics. This article provides an example in which the maximum likelihood estimator (MLE) of covariance parameters is still inconsistent under the framework of increasing domain asymptotics. Therefore, consistency may still be a problem under the framework of increasing domain asymptotics.