Asymptotic properties of estimators for random fields induced by stationary germ-grain models

Abstract

For a class of random fields, which is associated with stationary germ-grain models via conditionally bounded valuations, a mean value estimator is discussed. Its mean-square consistency and asymptotic normality is proven under certain conditions imposed on the dependence structure of the underlying point process and the grain distribution. For the Boolean model simple sufficient conditions on the grain distribution are derived, which guarantee the asymptotic normality using a central limit theorem for m-dependent random fields. For general germ-grain models beta-mixing conditions replace the assumption of m-dependence. For the asymptotic variance (or covariance matrix) three estimators are proposed and their mean-square consistency is shown under integrability conditions on mixed moments of the random fields. To evaluate estimators of the intrinsic volume densities of planar, stationary germ-grain models, computation algorithms are provided and discussed along some numerical examples for the Boolean model. Numerical results on running times, variability of the estimates, tests for normal distribution and significance tests for the vector of intrinsic volume densities complete the analysis.