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Large and moderate deviations for kernel-type estimators of the mean density of Boolean models. (English) Zbl 1390.62053

For applications in random geometry, generalizations of standard kernel-type estimators for random variables are considered for random closed sets with an integer Hausdorff dimension. Problems of this type occur e.g. in pattern recognition, image analysis, medicine, computer vision, material science. A broad class of random closed sets can be represented by Boolean models. A random closed set induces a random measure on the Borel \(\sigma\)-algebra on a Euclidean space \(\mathbb{R}^d\). The “mean density” is then defined by the derivative of the expected random measure with respect to the \(d\)-dimensional Hausdorff measure.
The aim of the present article is to further develop the estimation theory for mean densities in case of Boolean models: Large and moderate deviation principles for kernel-type estimators for mean densities are considered. Moreover, statistical properties, such as strong consistency and asymptotic confidence intervals, are studied. Some examples of Boolean models, such as the Boolean segment process, the Poisson point process and the Matérn cluster process are discussed.

MSC:

62G07 Density estimation
60F10 Large deviations
60D05 Geometric probability and stochastic geometry
62G20 Asymptotic properties of nonparametric inference
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[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. · Zbl 1253.60007
[3] Ambrosio, L., Fusco, N. and Pallara, D. (2000)., Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press, Oxford. · Zbl 0957.49001
[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. · Zbl 0822.60002
[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. · Zbl 1352.92076
[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. · Zbl 0797.60026
[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. · Zbl 1280.62042
[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. · Zbl 1305.62160
[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. · Zbl 1345.62050
[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. · Zbl 1330.62175
[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. · Zbl 1387.92047
[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. · Zbl 1143.92018
[18] Capasso, V. and Villa, E. (2006). On the continuity and absolute continuity of random closed sets., Stoch. An. Appl., 24, 381-397. · Zbl 1099.60010
[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. · Zbl 1154.60304
[20] Capasso, V. and Villa, E. (2008). On the geometric densities of random closed sets., Stoch. An. Appl., 26, 784-808. · Zbl 1151.60005
[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. · Zbl 1087.60001
[23] Chiu, S.N., Stoyan, D., Kendall, W.S. and Mecke, J. (2013)., Stochastic Geometry and its Applications, 3rd edition, John Wiley & Sons, Chichcester. · Zbl 1291.60005
[24] Daley, D.J. and Vere-Jones, D. (2003)., An introduction to the theory of point processes. Vol. I. 2nd edition. Springer, New York. · Zbl 1026.60061
[25] Daley, D.J. and Vere-Jones, D. (2008)., An introduction to the theory of point processes. Vol. II. 2nd edition. Springer, New York. · Zbl 1159.60003
[26] Dembo, A. and Zeitouni, O. (1998)., Large Deviations Techniques and Applications. 2nd edition, Springer. · Zbl 0896.60013
[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. · Zbl 0587.28004
[30] Federer, H. (1969)., Geometric Measure Theory. Springer, Berlin. · Zbl 0176.00801
[31] Gao, F. (2003). Moderate deviations and large deviations for kernel density estimators., J. Theoret. Probab., 16, 401-418. · Zbl 1041.62025
[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. · Zbl 1044.60006
[34] Kingman, J.F.C. (1993)., Poisson processes. Oxford University Press, Oxford. · Zbl 0771.60001
[35] Louani, D. (1998). Large deviations limit theorems for the kernel density estimator., Scand. J. Statist., 25, 243-253. · Zbl 0904.62060
[36] Matheron, G. (1975)., Random Sets and Integral Geometry. John Wiley & Sons, New York. · Zbl 0321.60009
[37] Molchanov, I. (1997)., Statistics of the Boolean Model for Practitioners and Mathematicians. Chichester, Wiley. · Zbl 0878.62068
[38] Molchanov, I. (2005)., Theory of Random Sets. Springer-Verlag, London. · Zbl 1109.60001
[39] Schneider, R. and Weil, W. (2008)., Stochastic and Integral Geometry. Springer-Verlag, Berlin Heidelberg. · Zbl 1175.60003
[40] Sheather, S.J. (2004). Density estimation, Statist. Sci., 19, 588-597. · Zbl 1100.62558
[41] Silverman, B.W. (1986)., Density Estimation for Statistics and Data Analysis. Chapman & Hall, London. · Zbl 0617.62042
[42] Szeliski, R. (2011)., Computer Vision. Algorithms and Applications. Springer-Verlag, London. · Zbl 1219.68009
[43] Villa E. (2010). Mean densities and spherical contact distribution function of inhomogeneous Boolean models., Stoch. An. Appl., 28, 480-504. · Zbl 1228.60023
[44] Villa E. (2014). On the local approximation of mean densities of random closed sets., Bernoulli, 20, 1-27. · Zbl 1291.60025
[45] Zähle, M. · Zbl 0575.60010
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