Variance estimation in compound decision theory under boundedness
Abstract
The normal means model is often studied under the assumption of a known variance. However, ignorance of the variance is a frequent issue in applications and basic theoretical questions still remain open in this setting. This article establishes that the sharp minimax rate of variance estimation in square error is $(\frac{\log\log n}{\log n})^2$ under arguably the most mild assumption imposed for identifiability: bounded means. The rate-optimal estimator proposed in this article achieves the optimal rate by estimating $O\left(\frac{\log n}{\log\log n}\right)$ cumulants and leveraging a variational representation of the noise variance in terms of the cumulants of the data distribution. The minimax lower bound involves a moment matching construction.
BibTeX
@inproceedings{
kotekal2024variance,
title={Variance estimation in compound decision theory under boundedness},
author={Subhodh Kotekal},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=HvCppnDykt}
}