Electronic Journal of Statistics

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

Federico Camerlenghi and Elena Villa

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1518750030

Digital Object Identifier
doi:10.1214/18-EJS1397

Mathematical Reviews number (MathSciNet)
MR3765603

Zentralblatt MATH identifier
06841010

Subjects
Primary: 62F12: Asymptotic properties of estimators 60F10: Large deviations 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Large deviations moderate deviations random closed sets confidence intervals stochastic geometry Boolean models

Rights
Creative Commons Attribution 4.0 International License.

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


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