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

Author

Listed:
  • Federico Camerlenghi

    (Università di Milano - Bicocca
    Collegio Carlo Alberto
    Bocconi University)

  • Claudio Macci

    (Università di Roma Tor Vergata)

  • Elena Villa

    (Università degli Studi di Milano)

Abstract

The mean density estimation of a random closed set in $$\mathbb {R}^d$$ 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 $$\Theta _n$$ Θ n belongs to the quite general class of stationary Boolean models with Hausdorff dimension $$n

Suggested Citation

  • Federico Camerlenghi & Claudio Macci & Elena Villa, 2021. "Asymptotic behavior of mean density estimators based on a single observation: the Boolean model case," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(5), pages 1011-1035, October.
    • Handle: RePEc:spr:aistmt:v:73:y:2021:i:5:d:10.1007_s10463-020-00775-y
    DOI: 10.1007/s10463-020-00775-y
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    References listed on IDEAS

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    1. Jiang, Tiefeng & Wang, Xiangchen & Rao, M. Bhaskara, 1992. "Moderate deviations for some weakly dependent random processes," Statistics & Probability Letters, Elsevier, vol. 15(1), pages 71-76, September.
    2. Burton, Robert M. & Dehling, Herold, 1990. "Large deviations for some weakly dependent random processes," Statistics & Probability Letters, Elsevier, vol. 9(5), pages 397-401, May.
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    4. Peter Diggle, 1985. "A Kernel Method for Smoothing Point Process Data," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 34(2), pages 138-147, June.
    5. Camerlenghi, F. & Capasso, V. & Villa, E., 2014. "On the estimation of the mean density of random closed sets," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 65-88.
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