Level sets of depth measures in abstract spaces

A. Cholaquidis (), R. Fraiman () and L. Moreno ()
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A. Cholaquidis: Universidad de la República
R. Fraiman: Universidad de la República
L. Moreno: Universidad de la República

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2023, vol. 32, issue 3, No 5, 942-957

Abstract: Abstract The lens depth of a point has been recently extended to general metric spaces, which is not the case for most depths. It is defined as the probability of being included in the intersection of two random balls centred at two random points X and Y, with the same radius d(X, Y). We prove that, on a separable and complete metric space, the level sets of the empirical lens depth based on an iid sample, converge in the Painlevé–Kuratowski sense, to its population counterpart. We also prove that, restricted to compact sets, the empirical level sets and their boundaries are consistent estimators, in Hausdorff distance, of their population counterparts, and analyse two real-life examples.

Keywords: Depth measures; Lens depth; Level sets; Metric spaces; Phylogenetic tree; 62R20; 62G35; 62G05 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s11749-023-00858-x

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