Mean Densities and Spherical Contact Distribution Function of Inhomogeneous Boolean Models

Optimal Bandwidth of the “Minkowski Content”-Based Estimator of the Mean Density of Random Closed Sets: Theoretical Results and Numerical Experiments

“…To lighten the presentation we shall use similar notation to previous works [9], [29], [30]; in particular, for the reader's convenience, we refer to [30,Section 2] (and references therein) for the mathematical background and more details on the Minkowski content notion and marked point process theory, while we refer to [9,Appendix] (and references therein) for the classical density estimation theory for random variables.…”

“…To this aim we shall have to distinguish two cases: the case in which the dimension n of Θ n equals d−1, and that one in which 0 < n < d−1, whereas the particular case n = 0 will be discussed in Section 4.2. In both cases the following lemma, which might be considered the generalization of Theorem 3.5 in [29] to product spaces, will be useful.…”

“…Indeed the study of the optimal bandwidth of the "Minkowski content"-based estimator has been left as an open problem in [29,Section 6] and in [30,Remark 14], and partially solved in [9,Section 4], where a formula is available only in the particular case in which the random set Θ n is a homogeneous Boolean model. So, this is the main goal of the present paper: to provide optimal bandwidths for the "Minkowski content"-based estimator for more general random closed sets, not necessarily homogeneous or Boolean models.…”

“…In literature different kinds of estimators are available, mostly for the homogeneous case. Recently the non homogeneous case has been faced by the authors; more precisely, two different kinds of estimators, asymptotically unbiased and weakly consistent, have been proposed: in [9] a kernel-type estimator generalizing the well-known kernel density estimator for random variables, and in [29] an estimator based on the notion of Minkowski content of a set. The study of the optimal bandwidth of the "Minkowski content"-based estimator has been left as an open problem in [29, Section 6] and in [30, Remark 14], and only partially solved in [9, Section 4], where a formula is available in the particular case of homogeneous Boolean models.…”

“…Throughout the paper we shall denote by Θ n any random closed set in R d with Hausdorff dimension n; we mean that a random closed set satisfies a certain property if it satisfies that property P-a.s. Recently, in [9], [29] and [30], two different kinds of estimators, asymptotically unbiased and weakly consistent, have been proposed by the authors: in [9] a kernel-type estimator generalizing the well-known kernel density estimator for random variables; in [29] and [30] (for Boolean models and more general germ-grain processes, respectively) an estimator based on the notion of Minkowski content. (See also [10] for a survey containing also some numerical experiments validating the theoretical results available so far).…”