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Peer-reviewedCC BYAnnals of the Institute of Statistical Mathematics

A lower bound for estimating Fréchet means

  • Shayan Hundrieser1
  • Benjamin Eltzner2
  • Stephan Huckemann2
  1. 1Department of Applied Mathematics, University of Twente, Drienerlolaan 5, 7522 NB, Enschede, The Netherlands
  2. 2Felix-Bernstein-Institute for Mathematical Statistics in the Biosciences, Georg-August-University Göttingen, Goldschmidtstraße 7, 37077, Göttingen, Germany
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Abstract

Fréchet means, conceptually appealing, generalize the Euclidean expectation to general metric spaces. We explore how well Fréchet means can be estimated from independent and identically distributed samples and uncover a fundamental limitation: In the vicinity of a probability distribution P with nonunique means, independent of sample size, it is not possible to uniformly estimate Fréchet means below a precision determined by the diameter of the set of Fréchet means of P . Our findings thus confirm inevitable statistical challenges in the estimation of Fréchet means on metric spaces for which there exist distributions with nonunique means. Illustrating the relevance of our lower bound, examples of extrinsic, intrinsic, Procrustes, diffusion and Wasserstein means showcase either deteriorating constants or slow convergence rates of empirical Fréchet means for samples near the regime of nonunique means.

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Cite

BibTeXRIS
BibTeX
@article{hundrieser2026lower,
  title = {{A lower bound for estimating Fréchet means}},
  author = {Shayan Hundrieser and Benjamin Eltzner and Stephan Huckemann},
  journal = {Annals of the Institute of Statistical Mathematics},
  year = {2026},
  month = jun,
  doi = {10.1007/s10463-026-00989-6},
  url = {https://doi.org/10.1007/s10463-026-00989-6},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - A lower bound for estimating Fréchet means
AU  - Shayan Hundrieser
AU  - Benjamin Eltzner
AU  - Stephan Huckemann
PY  - 2026
DA  - 2026/06/22
JO  - Annals of the Institute of Statistical Mathematics
DO  - 10.1007/s10463-026-00989-6
UR  - https://doi.org/10.1007/s10463-026-00989-6
AB  - Fréchet means, conceptually appealing, generalize the Euclidean expectation to general metric spaces. We explore how well Fréchet means can be estimated from independent and identically distributed samples and uncover a fundamental limitation: In the vicinity of a probability distribution P with nonunique means, independent of sample size, it is not possible to uniformly estimate Fréchet means below a precision determined by the diameter of the set of Fréchet means of P . Our findings thus confirm inevitable statistical challenges in the estimation of Fréchet means on metric spaces for which there exist distributions with nonunique means. Illustrating the relevance of our lower bound, examples of extrinsic, intrinsic, Procrustes, diffusion and Wasserstein means showcase either deteriorating constants or slow convergence rates of empirical Fréchet means for samples near the regime of nonunique means.
SN  - 0020-3157
SN  - 1572-9052
ER  -