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2018 Large and moderate deviations for kernel–type estimators of the mean density of Boolean models
Federico Camerlenghi, Elena Villa
Electron. J. Statist. 12(1): 427-460 (2018). DOI: 10.1214/18-EJS1397


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.


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Federico Camerlenghi. Elena Villa. "Large and moderate deviations for kernel–type estimators of the mean density of Boolean models." Electron. J. Statist. 12 (1) 427 - 460, 2018.


Received: 1 September 2017; Published: 2018
First available in Project Euclid: 16 February 2018

zbMATH: 06841010
MathSciNet: MR3765603
Digital Object Identifier: 10.1214/18-EJS1397

Primary: 60D05 , 60F10 , 62F12

Keywords: Boolean models , confidence intervals , large deviations , Moderate deviations , random closed sets , Stochastic geometry


Vol.12 • No. 1 • 2018
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