Asymptotic behavior of mean density estimators based on a single observation: the Boolean model case

Abstract

The mean density estimation of a random closed set in $$\mathbb {R}^d$$ , based on a single observation, is a crucial problem in several application areas. In the case of stationary random sets, a common practice to estimate the mean density is to take the n-dimensional volume fraction with observation window as large as possible. In the present paper, we provide large and moderate deviation results for these estimators when the random closed set $$\Theta _n$$ belongs to the quite general class of stationary Boolean models with Hausdorff dimension $$n<d$$ . Moreover, we establish a central limit theorem and a Berry–Esseen bound for the family of estimators under study. Our findings allow to recover some well-known results in the literature on Boolean models. Finally, we also provide a guideline for the estimation of the mean density of non-stationary Boolean models characterized by high intensity of the underlying Poisson point process.