  Information criteria, such as Akaike's information criterion and Bayesian
information criterion are often applied in model selection. However, their
asymptotic behaviors for selecting geostatistical regression models have not
been well studied, particularly under the fixed domain asymptotic framework
with more and more data observed in a bounded fixed region. In this article, we
study the generalized information criterion (GIC) for selecting geostatistical
regression models under a more general mixed domain asymptotic framework. Via
uniform convergence developments of some statistics, we establish the selection
consistency and the asymptotic loss efficiency of GIC under some regularity
conditions, regardless of whether the covariance model is correctly or wrongly
specified. We further provide specific examples with different types of
explanatory variables that satisfy the conditions. For example, in some
situations, GIC is selection consistent, even when some spatial covariance
parameters cannot be estimated consistently. On the other hand, GIC fails to
select the true polynomial order consistently under the fixed domain asymptotic
framework. Moreover, the growth rate of the domain and the degree of smoothness
of candidate regressors in space are shown to play key roles for model
selection.
