Adaptive Cauchy inference from invariant information
Abstract
Selecting weights and projection directions from the observations used for testing can invalidate a reference distribution. We develop a framework for exact conditional inference under rotation symmetry. A maximal invariant permits adaptation while leaving a uniform common phase for calibration. This yields explicit Cauchy references for invariantly selected weights and projections. Under separate block rotations, the same principle gives exact normal-score combinations with coefficients selected from all blockwise invariants. The calibration results require neither densities nor moments. To illustrate the framework, we study direction learning and power in spherical Gaussian location models, identifying a regime in which consistent detection is possible without consistent direction recovery. For heterogeneous functional data, a matched mean--residual construction produces the required null symmetry under a conditional Gaussian model. Simulations compare Radius and Blend, two representative procedures, with sample splitting, and an electrocardiogram analysis illustrates their application.
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BibTeX
@misc{liu2026adaptive,
title = {{Adaptive Cauchy inference from invariant information}},
author = {Yiming Liu and Shaochen Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16789},
eprint = {2610.16789},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16789},
}
RIS
TY - UNPB
TI - Adaptive Cauchy inference from invariant information
AU - Yiming Liu
AU - Shaochen Wang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16789
UR - https://arxiv.org/abs/2610.16789
AB - Selecting weights and projection directions from the observations used for testing can invalidate a reference distribution. We develop a framework for exact conditional inference under rotation symmetry. A maximal invariant permits adaptation while leaving a uniform common phase for calibration. This yields explicit Cauchy references for invariantly selected weights and projections. Under separate block rotations, the same principle gives exact normal-score combinations with coefficients selected from all blockwise invariants. The calibration results require neither densities nor moments. To illustrate the framework, we study direction learning and power in spherical Gaussian location models, identifying a regime in which consistent detection is possible without consistent direction recovery. For heterogeneous functional data, a matched mean--residual construction produces the required null symmetry under a conditional Gaussian model. Simulations compare Radius and Blend, two representative procedures, with sample splitting, and an electrocardiogram analysis illustrates their application.
ER -