Optimal Bandwidth of the “Minkowski Content”-Based Estimator of the Mean Density of Random Closed Sets: Theoretical Results and Numerical Experiments

The estimation of the mean density of random closed sets in $$\mathbb {R}^d$$^Rd with integer Hausdorff dimension $$n<d$$n<d is a problem of interest from both a theoretical and an applicative point of view. In literature different kinds of estimators are available, mostly for the homogeneous case. Recently the non-homogeneous case has been faced by the authors; more precisely, two different kinds of estimators, asymptotically unbiased and weakly consistent, have been proposed: in Camerlenghi et al. (J Multivar Anal 125:65–88, 2014) a kernel-type estimator generalizing the well-known kernel density estimator for random variables, and in Villa (Stoch Anal Appl 28:480–504, 2010) an estimator based on the notion of Minkowski content of a set. The study of the optimal bandwidth of the “Minkowski content”-based estimator has been left as an open problem in Villa (Stoch Anal Appl 28:480–504, 2010, Sect. 6) and in Villa (Bernoulli 20:1–27, 2014, Remark 14), and only partially solved in Camerlenghi et al. (J Multivar Anal 125:65–88, 2014, Sect. 4), where a formula is available in the particular case of homogeneous Boolean models. We give here a solution of such an open problem, by providing explicit formulas for the optimal bandwidth for quite general random closed sets (i.e., not necessarily Boolean models or homogeneous germ-grain models). We also discuss a series of relevant examples and corresponding numerical experiments to validate our theoretical results.