No abstract

Stationary random closed sets X in R d are considered whose realizations belong to the extended convex ring. A new approach is proposed to joint estimation of the specific intrinsic volumes V 0 ðXÞ; . . . ; V d ðXÞ of X; including the specific Euler-Poincare´characteristic V 0 ðXÞ; the specific surface area 2V dÀ1 ðXÞ; and the volume fraction V d ðXÞ of X: Nonparametric estimators are constructed, which can be represented by integrals of some stationary random fields. This implies in particular that these estimators are unbiased. Moreover, conditions are derived which ensure that they are mean-square consistent. A consistent estimator for their asymptotic covariance matrix is derived. r

No abstract

Conditions are derived for the asymptotic normality of a general class of vector-valued functionals of stationary Boolean models in the d-dimensional Euclidean space, where a Lindeberg-type central limit theorem for m-dependent random fields, m ∈ N, is applied. These functionals can be used to construct joint estimators for the vector of specific intrinsic volumes of the underlying Boolean model. Extensions to functionals of more general germ–grain models satisfying some mixing and integrability conditions are also discussed.