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PreprintCC BY 4.0arXiv

On the continuity of the Tukey depth function for fuzzy data

  • Luis González-De La Fuente
  • Alicia Nieto-Reyes
  • Pedro Terán
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Abstract

The practical use of statistical depth for fuzzy data requires regularity, inferential stability, and computational feasibility. This paper studies these aspects for the first time for the fuzzy depth, in particular for the fuzzy Tukey depth. In particular, we investigate the continuity of this depth with respect to each of its arguments. As a function of the elements of the underlying space, we prove its upper semicontinuity under the main metrics used in fuzzy spaces. As a function of the distribution with respect to which the depth is computed, we establish the almost sure uniform consistency of its empirical version for continuous fuzzy random variables. These results ensure closed depth regions and support the use of sample depth values to approximate their population counterparts. We also derive consistency of empirical deepest-point estimators and study a finite-grid approximation of the depth, supported by theoretical and simulation results.

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Cite

BibTeXRIS
BibTeX
@misc{fuente2026continuity,
  title = {{On the continuity of the Tukey depth function for fuzzy data}},
  author = {Luis González-De La Fuente and Alicia Nieto-Reyes and Pedro Terán},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18322},
  eprint = {2609.18322},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18322},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the continuity of the Tukey depth function for fuzzy data
AU  - Luis González-De La Fuente
AU  - Alicia Nieto-Reyes
AU  - Pedro Terán
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18322
UR  - https://arxiv.org/abs/2609.18322
AB  - The practical use of statistical depth for fuzzy data requires regularity, inferential stability, and computational feasibility. This paper studies these aspects for the first time for the fuzzy depth, in particular for the fuzzy Tukey depth. In particular, we investigate the continuity of this depth with respect to each of its arguments. As a function of the elements of the underlying space, we prove its upper semicontinuity under the main metrics used in fuzzy spaces. As a function of the distribution with respect to which the depth is computed, we establish the almost sure uniform consistency of its empirical version for continuous fuzzy random variables. These results ensure closed depth regions and support the use of sample depth values to approximate their population counterparts. We also derive consistency of empirical deepest-point estimators and study a finite-grid approximation of the depth, supported by theoretical and simulation results.
ER  -