Importance of sampling design in likelihood-based geostatistics

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Importance of sampling design in likelihood-based geostatistics

Ruben Roa

I hope that this forum is appropriate for the sort of topic i have in mind.

In Diggle, Ribeiro, and Christensen (An Introduction to Model-Based Geostatistics, April 15, 2002) the
sampling design is defined as the set of locations (the actual values and the labels), and in defining
their statistical model they include the disclaimer: "Note in particular that this model does not specify
the distribution of the sampling design, which as noted earlier is assumed to be independent of both [the
spatial process being estimated] and [the taking of measurements on this process]." Earlier they had written
"We shall assume either that the sampling design for [the locations] is deterministic or stochastic but
independent of the [spatial] process [to be estimated]."

I understand that the first disclaimer and the assumption thereby stated are unnecessary because when the
sampling design is defined as the set of locations, then it is deterministic by construction. This is because
all the locations in the spatial process to be estimated are known exactly (disregarding things such as GPS error)
so there is no distribution associated to them nor any subset of them. Maybe the probability distribution is associated not to the values of the locations but to the labels, but if the subset of locations is selected
purposively then the labels are also deterministic.

I think this is important because the restriction that the locations to be sampled should be independent
of the spatial process (as stated above in Diggle, Ribeiro and Christensen) is too strong, and reduces the scope
of likelihood-based geostatistics to only samples taken by scientists applying randomized or uniform designs.

Please correct me if i am wrong. Thanks.