Inference for Fréchet Mean Sets
Abstract
We construct confidence sets for the entire Fréchet mean set by inverting studentized pairwise risk inequalities. Gaussian multiplier calibration localized around empirical near minimizers gives asymptotic coverage at least the nominal level, uniformly over distribution classes satisfying empirical-process conditions. The construction requires neither unique ness nor a manifold structure. We characterize the limiting binding directions and give conditions for exact asymptotic coverage. On the intrinsic circle, exactness holds for every fixed non-point-mass law and uniformly over a family in which two means merge at a quar tic minimum. With ideal multiplier quantiles, the construction reproduces the Euclidean Wald set under both nonsingular and singular covariance. Under regular local conditions, it is first-order equivalent to intrinsic Wald inference.
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BibTeX
@misc{miyazaki2026inference,
title = {{Inference for Fréchet Mean Sets}},
author = {Ryunosuke Miyazaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18646},
eprint = {2610.18646},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18646},
}
RIS
TY - UNPB
TI - Inference for Fréchet Mean Sets
AU - Ryunosuke Miyazaki
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18646
UR - https://arxiv.org/abs/2610.18646
AB - We construct confidence sets for the entire Fréchet mean set by inverting studentized pairwise risk inequalities. Gaussian multiplier calibration localized around empirical near minimizers gives asymptotic coverage at least the nominal level, uniformly over distribution classes satisfying empirical-process conditions. The construction requires neither unique ness nor a manifold structure. We characterize the limiting binding directions and give conditions for exact asymptotic coverage. On the intrinsic circle, exactness holds for every fixed non-point-mass law and uniformly over a family in which two means merge at a quar tic minimum. With ideal multiplier quantiles, the construction reproduces the Euclidean Wald set under both nonsingular and singular covariance. Under regular local conditions, it is first-order equivalent to intrinsic Wald inference.
ER -