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

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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.

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Acknowledgements

The authors are grateful to an Associate Editor and the anonymous referees for their valuable comments and insightful suggestions, which led to a substantial improvement in the paper. All the authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Federico Camerlenghi gratefully acknowledges the financial support from the Italian Ministry of Education, University and Research (MIUR), “Dipartimenti di Eccellenza” Grant 2018–2022. Claudio Macci was partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C18000100006).

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Authors and Affiliations

  1. Dipartimento di Economia, Metodi Quantitativi e Strategie di Impresa, Università di Milano - Bicocca, Piazza dell’Ateneo Nuovo 1, 20126, Milan, Italy

    Federico Camerlenghi

  2. Collegio Carlo Alberto, Turin, Italy

    Federico Camerlenghi

  3. BIDSA, Bocconi University, Milan, Italy

    Federico Camerlenghi

  4. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133, Rome, Italy

    Claudio Macci

  5. Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133, Milan, Italy

    Elena Villa

Authors
  1. Federico Camerlenghi
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  2. Claudio Macci
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  3. Elena Villa
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Correspondence to Federico Camerlenghi.

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Camerlenghi, F., Macci, C. & Villa, E. Asymptotic behavior of mean density estimators based on a single observation: the Boolean model case. Ann Inst Stat Math 73, 1011–1035 (2021). https://doi.org/10.1007/s10463-020-00775-y

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  • DOI: https://doi.org/10.1007/s10463-020-00775-y

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