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**Spatial statistics.**
*(English)*
Zbl 1100.62574

Summary: When the distribution of \(\mathbf {X} \in \mathbb R^p\) depends only on its distance to some \(\theta_0 \in \mathbb R^p\), 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 \(\mathbf {X}\) and \(\theta_0\) are constrained to lie on the sphere in \(\mathbb R^p\). 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 \(U_i\) on one image and corresponding landmarks \(V_i\) 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 \(L_1\) and \(L_2\) registrations. Two principles are important when working with spatial statistics: (1) Assumptions, such as that the distribution of \(\mathbf {X}\) depends only on its distance to \(\theta_0\), 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 \(\Theta\) such as the sphere, techniques of elementary differential geometry can be used to minimize the distortions caused by using a coordinate system to reexpress \(\Theta\) in Euclidean parameters.

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