Electronic Journal of Statistics

Asymptotic results for multivariate estimators of the mean density of random closed sets

Federico Camerlenghi, Claudio Macci, and Elena Villa

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The problem of the evaluation and estimation of the mean density of random closed sets in $\mathbb{R} ^{d}$ with integer Hausdorff dimension $0<n<d$, is of great interest in many different scientific and technological fields. Among the estimators of the mean density available in literature, the so-called “Minkowski content”-based estimator reveals its benefits in applications in the non-stationary cases. We introduce here a multivariate version of such estimator, and we study its asymptotical properties by means of large and moderate deviation results. In particular we prove that the estimator is strongly consistent and asymptotically Normal. Furthermore we also provide confidence regions for the mean density of the involved random closed set in $m\geq1$ distinct points $x_{1},\ldots,x_{m}\in\mathbb{R} ^{d}$.

Article information

Electron. J. Statist., Volume 10, Number 2 (2016), 2066-2096.

Received: January 2016
First available in Project Euclid: 18 July 2016

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Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 60F10: Large deviations 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Minkowski content large deviations moderate deviations random closed sets confidence regions


Camerlenghi, Federico; Macci, Claudio; Villa, Elena. Asymptotic results for multivariate estimators of the mean density of random closed sets. Electron. J. Statist. 10 (2016), no. 2, 2066--2096. doi:10.1214/16-EJS1159. https://projecteuclid.org/euclid.ejs/1468849971

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  • [1] Aitchison J., Kay J.W., Lauder I.J. (2004)., Statistical Concepts and Applications in Clinical Medicine, Chapman and Hall/CRC, London.
  • [2] Ambrosio, L., Capasso, V., Villa, E. (2009). On the approximation of mean densities of random closed sets., Bernoulli 15, 1222–1242.
  • [3] Ambrosio, L., Fusco, N., Pallara, D. (2000)., Functions of Bounded Variation and Free Discontinuity Problems, Clarendon Press, Oxford.
  • [4] Baddeley, A., Barany, I., Schneider, R., Weil, W. (2007)., Stochastic Geometry. Lecture Notes in Mathematics, Vol. 1982. Springer, Berlin.
  • [5] Baddeley, A., Molchanov, I.S. (1997). On the expected measure of a random set. In:, Proceedings of the International Symposium on Advances in Theory and Applications of Random Sets (Fontainebleau, 1996), River Edge, NJ, World Sci. Publishing, 3–20.
  • [6] Bellettini, G. (2013)., Lecture notes on mean curvature flow, barriers and singular perturbations. Scuola Normale Superiore di Pisa (Nuova Serie), Vol. 12, Edizioni della Normale, Pisa.
  • [7] Billingsley, P. (1995)., Probability and Measure, 3rd edition. John Wiley & Sons.
  • [8] Camerlenghi, F., Capasso, V., Villa, E. (2014). On the estimation of the mean density of random closed sets., J. Multivariate Anal. 125, 65–88.
  • [9] Camerlenghi, F., Capasso, V., Villa, E. (2014). Numerical experiments for the estimation of mean densities of random sets. In: Proceedings of the 11th European Congress of Stereology and Image Analysis., Image Anal. Stereol. 33, 83–94.
  • [10] Camerlenghi, F., Villa, E. (2015). Optimal bandwidth of the “Minkowski content”-based estimator of the mean density of random closed sets: theoretical results and numerical experiments., J. Math. Imaging Vis. 125, 65–88.
  • [11] Capasso, V. (Ed.) (2003)., Mathematical Modelling for Polymer Processing. Polymerization, Crystallization, Manufacturing. ECMI Series on Mathematics in Industry, Vol. 2, Springer Verlag, Heidelberg.
  • [12] Capasso V., Dejana E., Micheletti A. (2008). Methods of stochastic geometry, and related statistical problems in the analysis and therapy of tumour growth and tumour-driven angiogenesis. In:, Selected Topics on Cancer Modelling, (N. Bellomo et al. Eds.), Birkhauser, Boston, 299–335.
  • [13] Capasso V., Micheletti A. (2006). Stochastic geometry and related statistical problems in biomedicine. In:, Complex System in Biomedicine, (A. Quarteroni et al. Eds.), Springer, Milano, 35–69.
  • [14] Capasso V., Micheletti A., Morale D. (2008). Stochastic geometric models and related statistical issues in tumour-induced angiogenesis., Mathematical Biosciences, 214, 20–31.
  • [15] Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J. (2013)., Stochastic Geometry and its Applications, 3rd edition. John Wiley & Sons, Chichcester.
  • [16] Daley, D.J., Vere-Jones, D. (1988)., An Introduction to the Theory of Point Processes. Springer Series in Statistics, New York.
  • [17] Dembo, A., Zeitouni, O. (1998)., Large Deviations Techniques and Applications, 2nd edition. Springer.
  • [18] Devroye L., Györfi L., Lugosi G. (1996)., A Probabilistic Theory of Pattern Recognition, Springer Series in Stochastic Modelling and Applied Probability, New York.
  • [19] Heinrich, L. (2005). Large deviations of the empirical volume fraction for stationary Poisson grain models., Ann. Appl. Probab. 15(1A), 392–420.
  • [20] Jacod, J., Protter, P. (2003)., Probability essentials, 2nd edition. Springer-Verlag, Berlin.
  • [21] Karr, A.F. (1986)., Point Processes and Their Statistical Inference. Marcel Dekker, New York.
  • [22] Matheron, G. (1975)., Random Sets and Integral Geometry. John Wiley & Sons, New York.
  • [23] Sheather, S.J. (2004). Density estimation., Statistical Science, 19, 588–597.
  • [24] Szeliski, R. (2011)., Computer Vision. Algorithms and Applications. Springer-Verlag, London.
  • [25] Villa, E. (2010). Mean densities and spherical contact distribution function of inhomogeneous Boolean models., Stoch. An. Appl. 28, 480–504.
  • [26] Villa, E. (2014). On the local approximation of mean densities of random closed sets., Bernoulli 20, 1–27.
  • [27] Zähle, M. (1982). Random processes of Hausdorff rectifiable closed sets., Math. Nachr. 108, 49–72.