## 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.

### MSC:

 62H11 Directional data; spatial statistics 62H35 Image analysis in multivariate analysis
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### References:

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