On the estimation of the mean density of random closed sets

Many real phenomena may be modeled as random closed sets in Rd, of different Hausdorff dimensions. Of particular interest are cases in which their Hausdorff dimension, say n, is strictly less than d, such as fiber processes, boundaries of germ–grain models, and n-facets of random tessellations. A crucial problem is the estimation of pointwise mean densities of absolutely continuous, and spatially inhomogeneous random sets, as defined by the authors in a series of recent papers. While the case n = 0 (random vectors, point processes, etc.) has been, and still is, the subject of extensive literature, in this paper we face the general case of any n < d; pointwise density estimators which extend the notion of kernel density estimators for random vectors are analyzed, together with a previously proposed estimator based on the notion of Minkowski content. In a series of papers, the authors have established the mathematical framework for obtaining suitable approximations of such mean densities. Here we study the unbiasedness and consistency properties, and identify optimal bandwidths for all proposed estimators, under sufficient regularity conditions. We show how some known results in the literature follow as particular cases. A series of examples throughout the paper, both non-stationary, and stationary, are provided to illustrate various relevant situations.