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 standpoint. Nowadays different kinds of estimators are available in the literature, in of the mean density is a problem of interest both from a theoretical and computational 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.

Large and moderate deviations for kernel- type estimators of the mean density of Boolean models / F. Camerlenghi, E. Villa. - In: ELECTRONIC JOURNAL OF STATISTICS. - ISSN 1935-7524. - 12:1(2018), pp. 427-460. [10.1214/18-EJS1397]

Large and moderate deviations for kernel-€“type estimators of the mean density of Boolean models

E. Villa
2018

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 standpoint. Nowadays different kinds of estimators are available in the literature, in of the mean density is a problem of interest both from a theoretical and computational 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.
Boolean models; Confidence intervals; Large deviations; Moderate deviations; Random closed sets; Stochastic geometry; Statistics and Probability
Settore MAT/06 - Probabilita' e Statistica Matematica
ELECTRONIC JOURNAL OF STATISTICS
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/560503
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