## Electronic Journal of Statistics

### Large and moderate deviations for kernel–type estimators of the mean density of Boolean models

#### Abstract

The mean density of a random closed set with integer Hausdorff dimension is a crucial notion in stochastic geometry, in fact it is a fundamental tool in a large variety of applied problems, such as image analysis, medicine, computer vision, etc. Hence the estimation of the mean density is a problem of interest both from a theoretical and computational standpoint. Nowadays different kinds of estimators are available in the literature, in particular here we focus on a kernel–type estimator, which may be considered as a generalization of the traditional kernel density estimator of random variables to the case of random closed sets. The aim of the present paper is to provide asymptotic properties of such an estimator in the context of Boolean models, which are a broad class of random closed sets. More precisely we are able to prove large and moderate deviation principles, which allow us to derive the strong consistency of the estimator of the mean density as well as asymptotic confidence intervals. Finally we underline the connection of our theoretical findings with classical literature concerning density estimation of random variables.

#### Article information

Source
Electron. J. Statist., Volume 12, Number 1 (2018), 427-460.

Dates
First available in Project Euclid: 16 February 2018

https://projecteuclid.org/euclid.ejs/1518750030

Digital Object Identifier
doi:10.1214/18-EJS1397

Mathematical Reviews number (MathSciNet)
MR3765603

Zentralblatt MATH identifier
06841010

#### Citation

Camerlenghi, Federico; Villa, Elena. Large and moderate deviations for kernel–type estimators of the mean density of Boolean models. Electron. J. Statist. 12 (2018), no. 1, 427--460. doi:10.1214/18-EJS1397. https://projecteuclid.org/euclid.ejs/1518750030

#### References

• [1] Aitchison, J., Kay, J.W. and Lauder, I.J. (2004)., Statistical Concepts and Applications in Clinical Medicine. Chapman and Hall/CRC, London.
• [2] Ambrosio, L., Capasso, V. and Villa, E. (2009). On the approximation of mean densities of random closed sets., Bernoulli, 15, 1222–1242.
• [3] Ambrosio, L., Fusco, N. and Pallara, D. (2000)., Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford.
• [4] Baddeley, A., Barany, I., Schneider, R. and Weil, W. (2007)., Stochastic Geometry. Lecture Notes in Mathematics 1982, Springer, Berlin.
• [5] Baddeley, A. and 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] Beneš, V. and Rataj, J. (2004)., Stochastic Geometry: Selected Topics. Kluwer, Dordrecht.
• [7] Billingsley, P. (1995)., Probability and Measure, 3rd edition. John Wiley & Sons.
• [8] Bonilla, L.L., Capasso, V., Alvaro, M., Carretero, and Terragni, F. (2017). On the mathematical modelling of tumor–induced angiogenesis., Math. Biosci. Eng., 14, 45–66.
• [9] Bryc, W. (1993) A remark on the connection between the large deviation principle and the central limit theorem., Statist. Probab. Lett., 18, 253–256.
• [10] Camerlenghi, F., Capasso, V. and Villa, E. (2014). On the estimation of the mean density of random closed sets., J. Multivariate Anal., 125, 65–88.
• [11] Camerlenghi, F., Capasso, V. and 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.
• [12] Camerlenghi, F., Macci, C. and Villa, E. (2016). Asymptotic results for the estimation of the mean density of random closed sets., Electron. J. Stat., 10, 2066–2096.
• [13] Camerlenghi, F. and 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 Vision, 53, 264–287.
• [14] Mathematical Modelling for Polymer Processing. Polymerization, Crystallization, Manufacturing (V. Capasso, Editor). ECMI Series on Mathematics in Industry Vol 2, Springer Verlag, Heidelberg, 2003.
• [15] Capasso, V., Dejana, E. and 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.
• [16] Capasso, V. and 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.
• [17] Capasso, V., Micheletti, A. and Morale, D. (2008), Stochastic geometric models and related statistical issues in tumour-induced angiogenesis., Math. Biosci., 214, 20–31.
• [18] Capasso, V. and Villa, E. (2006). On the continuity and absolute continuity of random closed sets., Stoch. An. Appl., 24, 381–397.
• [19] Capasso, V. and Villa, E. (2007). On mean densities of inhomogeneous geometric processes arising in material sciences and medicine., Image Anal. Setreol., 26, 23–36.
• [20] Capasso, V. and Villa, E. (2008). On the geometric densities of random closed sets., Stoch. An. Appl., 26, 784–808.
• [21] Charalambides, C.A. (2002)., Enumerative combinatorics. CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL.
• [22] Charalambides, C.A. (2005)., Combinatorial methods in discrete distributions. Hoboken, NJ: Wiley.
• [23] Chiu, S.N., Stoyan, D., Kendall, W.S. and Mecke, J. (2013)., Stochastic Geometry and its Applications, 3rd edition, John Wiley & Sons, Chichcester.
• [24] Daley, D.J. and Vere-Jones, D. (2003)., An introduction to the theory of point processes. Vol. I. 2nd edition. Springer, New York.
• [25] Daley, D.J. and Vere-Jones, D. (2008)., An introduction to the theory of point processes. Vol. II. 2nd edition. Springer, New York.
• [26] Dembo, A. and Zeitouni, O. (1998)., Large Deviations Techniques and Applications. 2nd edition, Springer.
• [27] Devroye, L. and Györfi, L. (1985)., Nonparametric density estimation: the $L_1$ view. Wiley, New York.
• [28] Devroye, L., Györfi, L. and Lugosi, G. (1996)., A Probabilistic Theory of Pattern Recognition. Springer Series in Stochastic Modelling and Applied Probability, New York.
• [29] Falconer, K.J. (1986)., The Geometry of Fractal Sets. Cambridge University Press, Cambridge.
• [30] Federer, H. (1969)., Geometric Measure Theory. Springer, Berlin.
• [31] Gao, F. (2003). Moderate deviations and large deviations for kernel density estimators., J. Theoret. Probab., 16, 401–418.
• [32] Härdle, W. (1991)., Smoothing Techniques with Implementation in S. Springer-Verlag, New York.
• [33] Hug, D. and Last, G. (2000). On support measures in Minkowski spaces and contact distributions in stochastic geometry., Ann. Prob., 28, 796–850.
• [34] Kingman, J.F.C. (1993)., Poisson processes. Oxford University Press, Oxford.
• [35] Louani, D. (1998). Large deviations limit theorems for the kernel density estimator., Scand. J. Statist., 25, 243–253.
• [36] Matheron, G. (1975)., Random Sets and Integral Geometry. John Wiley & Sons, New York.
• [37] Molchanov, I. (1997)., Statistics of the Boolean Model for Practitioners and Mathematicians. Chichester, Wiley.
• [38] Molchanov, I. (2005)., Theory of Random Sets. Springer-Verlag, London.
• [39] Schneider, R. and Weil, W. (2008)., Stochastic and Integral Geometry. Springer-Verlag, Berlin Heidelberg.
• [40] Sheather, S.J. (2004). Density estimation, Statist. Sci., 19, 588–597.
• [41] Silverman, B.W. (1986)., Density Estimation for Statistics and Data Analysis. Chapman & Hall, London.
• [42] Szeliski, R. (2011)., Computer Vision. Algorithms and Applications. Springer-Verlag, London.
• [43] Villa E. (2010). Mean densities and spherical contact distribution function of inhomogeneous Boolean models., Stoch. An. Appl., 28, 480–504.
• [44] Villa E. (2014). On the local approximation of mean densities of random closed sets., Bernoulli, 20, 1–27.
• [45] Zähle, M. (1982). Random processes of Hausdorff rectifiable closed sets., Math. Nachr., 108, 49–72.