Estimation of the intensity of stationary flat processes

The intensity of a stationary process of k-dimensional affine subspaces (k-flats) of d with directional distribution from a given family R is estimated by observing the process in a compact window. To this end we introduce a type of unbiased estimator (the R-estimator) using the available information about the directional distribution. Special cases are estimators for the intensity of stationary k-flat processes (1) with known directional distribution, (2) with directional distribution invariant with respect to a subgroup of the group of rotations in d and (3) with unknown directional distribution. We give sufficient conditions for the R-estimator to be the uniformly best unbiased estimator for the intensity of stationary Poisson k-flat processes with directional distribution in R. Equivalent statements for certain types of stationary Cox flat processes can be deduced directly from the results in the Poisson case. Moreover, we consider stationary ergodic flat processes with directional distribution in R and general stationary flat processes with unknown directional distribution, all with a non-degeneracy property. In both cases our estimator turns out to be the uniformly best unbiased estimator from a restricted set of estimators. The result for general stationary flat processes is proved with the help of a factorization result for the second factorial moment measure.