  Modeling spatial overdispersion requires point processes models with finite
dimensional distributions that are overdisperse relative to the Poisson.
Fitting such models usually heavily relies on the properties of stationarity,
ergodicity, and orderliness. And, though processes based on negative binomial
finite dimensional distributions have been widely considered, they typically
fail to simultaneously satisfy the three required properties for fitting.
Indeed, it has been conjectured by Diggle and Milne that no negative binomial
model can satisfy all three properties. In light of this, we change
perspective, and construct a new process based on a different overdisperse
count model, the Generalized Waring Distribution. While comparably tractable
and flexible to negative binomial processes, the Generalized Waring process is
shown to possess all required properties, and additionally span the negative
binomial and Poisson processes as limiting cases. In this sense, the GW process
provides an approximate resolution to the conundrum highlighted by Diggle and
Milne.
