The mean density estimation of a random closed set in Rd, 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 Θ 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.

Asymptotic behavior of mean density estimators based on a single observation : the Boolean model case / F. Camerlenghi, C. Macci, E. Villa. - In: ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS. - ISSN 0020-3157. - 73:5(2021), pp. 1011-1035. [10.1007/s10463-020-00775-y]

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

E. Villa
2021

Abstract

The mean density estimation of a random closed set in Rd, 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 Θ 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.
Hausdorff measure; Large deviations; Moderate deviations; Point processes; Stochastic geometry
Settore MAT/06 - Probabilita' e Statistica Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/867097
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