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.


62G07 Density estimation
60F10 Large deviations
60D05 Geometric probability and stochastic geometry
62G20 Asymptotic properties of nonparametric inference
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