Joint estimators for the specific intrinsic volumes of stationary random sets

Stationary random closed sets Ξ 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 ( Ξ ) , … , V ¯ d ( Ξ ) of Ξ , including the specific Euler–Poincaré characteristic V ¯ 0 ( Ξ ) , the specific surface area 2 V ¯ d - 1 ( Ξ ) , and the volume fraction V ¯ d ( Ξ ) of Ξ . 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.