Approximately optimal spatial design: How good is it?

The increasing recognition of the association between adverse human health conditions and many environmental substances as well as processes has led to the need to monitor them. An important problem that arises in environmental statistics is the design of the locations of the monitoring stations for those environmental processes of interest. One particular design criterion for monitoring networks that tries to reduce the uncertainty about predictions of unseen processes is called the maximum-entropy design. However, this design criterion involves a hard optimization problem that is computationally intractable for large data sets. Previous work of Wang et al. (2017) examined a probabilistic model that can be implemented efficiently to approximate the underlying optimization problem. In this paper, we attempt to establish statistically sound tools for assessing the quality of the approximations.