On the Estimation of Integrated Covariance Functions of Stationary Random Fields

For stationary vector-valued random fields on <formula format="inline"><file name="sjos_663_mu1.gif" type="gif" /></formula> the asymptotic covariance matrix for estimators of the mean vector can be given by integrated covariance functions. To construct asymptotic confidence intervals and significance tests for the mean vector, non-parametric estimators of these integrated covariance functions are required. Integrability conditions are derived under which the estimators of the covariance matrix are mean-square consistent. For random fields induced by stationary Boolean models with convex grains, these conditions are expressed by sufficient assumptions on the grain distribution. Performance issues are discussed by means of numerical examples for Gaussian random fields and the intrinsic volume densities of planar Boolean models with uniformly bounded grains. Copyright (c) 2009 Board of the Foundation of the Scandinavian Journal of Statistics.