Level sets of depth measures in abstract spaces Level sets of depth measures in abstract spaces

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...

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Bibliographic Details
Published in:Test (Madrid, Spain) Vol. 32; no. 3; pp. 942 - 957
Main Authors: Cholaquidis, A., Fraiman, R., Moreno, L.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2023
Subjects:
Economics
Finance
Insurance
Management
Mathematics and Statistics
Original Paper
Statistical Theory and Methods
Statistics
Statistics for Business
62G05
62R20
Lens depth
Phylogenetic tree
Metric spaces
Level sets
Depth measures
62G35
ISSN:1133-0686, 1863-8260
Online Access:Get full text
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Summary: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.
Bibliography:ObjectType-Article-1
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content type line 23
ISSN:1133-0686
1863-8260
DOI:10.1007/s11749-023-00858-x