Molchanov, Ilya S. Statistics of the Boolean model: From the estimation of means to the estimation of distributions. (English) Zbl 0834.62096 Adv. Appl. Probab. 27, No. 1, 63-86 (1995). 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) Cited in 19 Documents 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 Keywords:grain distribution; Boolean model; stationary random closed sets; stationary Poisson process of compact sets; bounded sampling window; tangent point; marked point process; central limit theorem; surface area measure; asymptotic normality PDFBibTeX XMLCite \textit{I. S. Molchanov}, Adv. Appl. Probab. 27, No. 1, 63--86 (1995; Zbl 0834.62096) Full Text: DOI