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.
Highlights
The mean density of lower dimensional random closed sets, such as fiber processes and surfaces of full dimensional random sets, is an important quantity which arises in different scientific fields
The mean density of a random closed set with integer Hausdorff dimension is a crucial notion in stochastic geometry, it is a fundamental tool in a large variety of applied problems, such as image analysis, medicine, computer vision, etc
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
Summary
The mean density of lower dimensional random closed sets, such as fiber processes and surfaces of full dimensional random sets, is an important quantity which arises in different scientific fields. The main aim of the present paper is the investigation of large and moderate deviation principles of λκΘ,nN (x) for a large class of random closed sets, known as Boolean models, leaving to subsequent works extensions to more general classes. The present paper may be seen as the first step in extending large and moderate deviation principles for kernel density estimators of random variables to the case of kernel-type estimators of the mean density of random sets. Large and moderate deviation principles for the kernel–type estimator of the mean density are presented, namely in Theorem 2 and Theorem 3, respectively. These theorems are the basic building blocks to derive statistical properties of such an estimator. For the reader’s convenience, the proofs of the main theorems, and some related technical lemmas, are deferred to Appendix A
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