  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.
