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Level sets of depth measures in abstract spaces

Author

Listed:
  • A. Cholaquidis

    (Universidad de la República)

  • R. Fraiman

    (Universidad de la República)

  • L. Moreno

    (Universidad de la República)

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.

Suggested Citation

  • A. Cholaquidis & R. Fraiman & L. Moreno, 2023. "Level sets of depth measures in abstract spaces," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(3), pages 942-957, September.
  • Handle: RePEc:spr:testjl:v:32:y:2023:i:3:d:10.1007_s11749-023-00858-x
    DOI: 10.1007/s11749-023-00858-x
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    References listed on IDEAS

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    1. Gerda Claeskens & Mia Hubert & Leen Slaets & Kaveh Vakili, 2014. "Multivariate Functional Halfspace Depth," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 411-423, March.
    2. López-Pintado, Sara & Romo, Juan, 2009. "On the Concept of Depth for Functional Data," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 718-734.
    3. Agostinelli, Claudio, 2018. "Local half-region depth for functional data," Journal of Multivariate Analysis, Elsevier, vol. 163(C), pages 67-79.
    4. Robert Serfling, 2002. "Quantile functions for multivariate analysis: approaches and applications," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(2), pages 214-232, May.
    5. Ruts, Ida & Rousseeuw, Peter J., 1996. "Computing depth contours of bivariate point clouds," Computational Statistics & Data Analysis, Elsevier, vol. 23(1), pages 153-168, November.
    6. Fraiman, Ricardo & Gamboa, Fabrice & Moreno, Leonardo, 2019. "Connecting pairwise geodesic spheres by depth: DCOPS," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 81-94.
    7. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    8. Ilya S. Molchanov, 1998. "A Limit Theorem for Solutions of Inequalities," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 25(1), pages 235-242, March.
    9. Amy Willis, 2019. "Confidence Sets for Phylogenetic Trees," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(525), pages 235-244, January.
    10. Ricardo Fraiman & Graciela Muniz, 2001. "Trimmed means for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 10(2), pages 419-440, December.
    11. D. Barden & H. Le & M. Owen, 2018. "Limiting behaviour of Fréchet means in the space of phylogenetic trees," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 70(1), pages 99-129, February.
    12. Zhenyu Liu & Reza Modarres, 2011. "Lens data depth and median," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(4), pages 1063-1074.
    13. Cuevas, Antonio & Fraiman, Ricardo, 2009. "On depth measures and dual statistics. A methodology for dealing with general data," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 753-766, April.
    14. Dai, Wenlin & Genton, Marc G., 2019. "Directional outlyingness for multivariate functional data," Computational Statistics & Data Analysis, Elsevier, vol. 131(C), pages 50-65.
    15. Karl Mosler & Pavlo Mozharovskyi, 2017. "Fast DD-classification of functional data," Statistical Papers, Springer, vol. 58(4), pages 1055-1089, December.
    16. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2017. "Multivariate and functional classification using depth and distance," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 11(3), pages 445-466, September.
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