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Statistics of the Boolean model: From the estimation of means to the estimation of distributions. (English) Zbl 0834.62096

The Boolean model \(\Theta\) is the most important model for stationary random closed sets in \(\mathbb{R}^d\). It arises as the union set of a stationary Poisson process of compact sets. The latter is characterized by the intensity \(\lambda\) of its germs and by the distribution \(P_0\) of the typical grain \(\Theta_0\). The paper is devoted to the statistical problem of estimating \(\lambda\) and \(P_0\) on the basis of observations of \(\Theta\) in a bounded sampling window.
The basic assumption is that the grains are convex since this allows to define, for a given direction \(u\) and each grain \(F\), a tangent point \(n_u (F)\) of \(F\) in direction \(u\) (a boundary point with outer normal \(-u\)). The Boolean model \(\Theta\) then determines a point process \(N^+ (u)\) in \(\mathbb{R}^d\) consisting of the tangent points \(n_u (\Theta_i)\) (of the grains \(\Theta_i\)) which are not covered by other grains. More generally, for directions \(u_1, \dots, u_m\), \(N^+ (u_1, \dots, u_m)\) is the marked point process of uncovered tangent points in directions \(u_1, \dots, u_n\). It is well-known that \(\lambda\) can be estimated from the intensity of \(N^+ (u)\).
In the first part of the paper, it is shown that the second- and higher- order characteristics of \(N^+ (u)\) and/or \(N^+ (u_1, \dots, u_m)\) can be used to estimate not only \(\lambda\) but also the distribution \(P_0\). For an improved estimator of \(\lambda\), a central limit theorem is proved.
In the second part, the surface area measure of a Boolean model in an open window is considered. The second moment measure of this random measure is given and used to derive a central limit theorem for the surface area measure. Finally, the results are used to define a strongly consistent estimator for the expected surface area measure of the primary grain and to show its asymptotic normality.
Reviewer: W.Weil (Karlsruhe)

MSC:

62M30 Inference from spatial processes
60D05 Geometric probability and stochastic geometry
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
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