When the distribution of X∈ℝp depends only on its distance to some θ0∈ℝp, we discuss results from Hössjer and Croux and Neeman and Chang on rank score statistics. Similar results from Neeman and Chang are also given when X and θ0 are constrained to lie on the sphere in ℝp. Results from Ko and Chang on M estimation for spatial models in Euclidean space and the sphere are also discussed. Finally we discuss a regression type model: the image registration problem. We have landmarks ui on one image and corresponding landmarks Vi on a second image. It is desired to bring the two images into closest coincidence through a translation, rotation and scale change. The techniques and principles of this paper are summarized through extensive discussion of an example in three-dimensional image registration and a comparison of the L1 and L2 registrations. Two principles are important when working with spatial statistics: (1) Assumptions, such as that the distribution of X depends only on its distance to θ0, introduce symmetries to spatial models which, if properly used, greatly simplify statistical calculations. These symmetries can be expressed in a more general setting by using the notion of statistical group models. (2) When working with a non-Euclidean parameter space Θ such as the sphere, techniques of elementary differential geometry can be used to minimize the distortions caused by using a coordinate system to reexpress Θ in Euclidean parameters.
"Spatial Statistics." Statist. Sci. 19 (4) 624 - 635, November 2004. https://doi.org/10.1214/088342304000000567