Exotic areas of statistics, such as spatial statistics in general and spatial point processes in particular often recapitulate the history of statistics. The first formal inference is nonparametric, using method of moments estimators. At this stage there is no modelling. Ordinary statistics was in this phase a century ago with Pearson families of curves fit by method of moments. Time series was in this phase in the 1950s with spectral analysis. Spatial statistics was in this phase in the 1970s. Ripley's K-functions are an example. So is spatial autocorrelation analysis of lattice processes.
The next phase involves the introduction of parametric statistical models, efficient estimation methods, and hypothesis tests, which happened in ordinary statistics in the 1920s and 1930s, in time series in the 1960s, and is only now happening in spatial statistics. As this phase is entered, there are many special methods developed that are later replaced by the methods commonplace in ordinary statistics. An example of this would be the replacement of pseudolikelihood by ordinary likelihood, which is not yet complete.
Some readers, no doubt, will object to this story. Spatial statistics, they will say, is special and needs its own special methods. This is partly true. Aspects of spatial statistics that have no analogue in ordinary statistics must use special methods, but aspects that are universal, stochastic modelling and estimation of parameters are not without their analogues in ordinary statistics. It stands to reason that what works well in general should also work well in spatial statistics. If my work in spatial statistics has any clear goal, it is to make spatial statistics as much like ordinary statistics as possible. I hope readers will agree that this chapter shows we are advancing in that direction.