研究高维均值检验在任意截断下的极限，揭示了隐藏概率质量如何影响检测能力。
Mean Testing under Truncation beyond Gaussian
- 提出基于方向矩的截断偏差分析，给出偏差上界为O(ν_{P,p}ε^{1-1/p})
- 发现信号强度低于阈值时，无论数据多大都无法区分假设，存在信息论下限
- 在中位数正则条件下，偏差降至O(ε)，测试样本复杂度恢复到经典√d量级
我们刻画了在任意截断下高维均值检验的根本极限，其中样本来自未知截断集S的条件分布，该集合可能隐藏最多ε比例的概率质量。对于具有p阶方向矩大小不超过ν_{P,p}的分布，截断引起的偏差为O(ν_{P,p}ε^{1-1/p})。这一偏差导致一个尖锐的信息论可检测性下限：当信号α低于此阈值时，即使有无限数据，零假设与备择假设也不可区分。在此阈值以上，我们证明了一种简单的二阶检验方法，其样本复杂度接近最优，为n = O(‖Σ_P‖ / (α - 4ν_{P,p}ε^{1-1/p})² √d)。此外，我们识别出一种结构上的突破：在方向中位数正则性假设下，截断偏差可优化至线性阶O(ε)。这揭示了一个中间区域——估计需Θ(d)样本实现均匀恢复，而测试在消除截断偏差后可恢复经典Θ(√d)速率。我们的结果提供了一个统一框架，连接有限矩、次高斯及中位数正则结构情形下的均值检验。
原文摘要 · Abstract (English)
We characterize the fundamental limits of high-dimensional mean testing under arbitrary truncation, where samples are drawn from the conditional distribution $P(\cdot \mid S)$ for an unknown truncation set $S$ that may hide up to an $\varepsilon$-fraction of the probability mass. For distributions with $p$-th directional moments of magnitude at most $ν_{P,p}$, truncation induces a bias of order $O(ν_{P,p}\varepsilon^{1-1/p})$. This bias creates a sharp information-theoretic detectability floor: when the signal $α$ falls below this threshold, the null and alternative hypotheses are indistinguishable even with infinite data. Above this floor, we prove that a simple second-order test achieving near-optimal sample complexity $n = O\!\left(\frac{\|Σ_P\|}{(α-4ν_{P,p}\varepsilon^{1-1/p})^2}\sqrt{d}\right)$. We further identify a structural escape from this finite-moment bias barrier. Under a directional median regularity assumption, truncation bias improves to linear order $O(\varepsilon)$. This reveals an intermediate regime in which estimation requires $Θ(d)$ samples for uniform recovery, while testing recovers the classical $Θ(\sqrt d)$ rate once truncation bias is eliminated. Together, our results provide a unified framework for mean testing under truncation, connecting finite-moment, sub-Gaussian, and median-regular structural regimes.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。