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arXiv 2604.26992math.STstat.MEstat.MLstat.TH
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在Efron的高斯双组模型中适应性鲁棒置信区间

Adaptive Robust Confidence Intervals in Efron's Gaussian Two-Groups Model

University of Chicago（芝加哥大学）；Northwestern University（西北大学）

Qiaosen Wang, Shuwen Chai, Chao Gao

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AI总结：

研究在Efron的高斯双组模型中，针对未知污染比例的鲁棒置信区间，发现其长度比已知污染比例时更差，提出基于傅里叶变换的认证方法。

摘要：

鲁棒不确定性量化在现代数据分析中越来越重要，通常在Huber模型下形式化，允许ε比例的任意破坏。然而，在许多实验科学中，测量协议受控，污染更可能来自上游。受这种噪声无关性质的启发，我们研究Efron高斯双组模型中空位置参数θ的置信区间，其中未知比例ε的观测具有任意偏移均值，但所有样本共享相同的加性高斯测量噪声分布。我们表征了在给定覆盖水平下，置信区间的最小最大长度，均匀覆盖未知污染比例和所有噪声无关对手。尽管先前工作表明，当ε未知时，θ的最小最大点估计率不恶化，但我们的结果揭示，给定σ²时，适应未知ε的置信区间的最小最大长度为σ(n^{-1/4}+ε^{1/2}/max{1, log(en ε²)}^{1/2})，比ε已知时的最优长度差多项式。当σ²也未知时，我们显示进一步退化：没有适应性鲁棒置信区间可以比Ω(σn^{-1/8})更短。算法上，我们引入基于Carathéodory正半定性约束的傅里叶认证过程。通过扫描候选点并接受残差特征函数可证实与高斯位置混合一致的点，我们的算法在已知方差设置中达到最小最大下界，并在多项式时间内可计算。

英文摘要：

Robust uncertainty quantification is increasingly important in modern data analysis and is often formalized under Huber's model, which allows an $\varepsilon$-fraction of arbitrary corruptions. In many experimental sciences, however, the measurement protocol is well controlled, and contamination is more plausibly introduced upstream. Motivated by this noise-oblivious nature of adversaries, we study confidence intervals for the null location parameter $θ$ in Efron's Gaussian two-groups model, where an unknown fraction $\varepsilon$ of observations have arbitrarily shifted means, but all samples share the same law of additive Gaussian measurement noise with variance $σ^2$. We characterize the minimax-optimal length among confidence intervals with a prescribed coverage level uniformly over the unknown contamination proportion and all noise-oblivious adversaries. Although prior work has shown that the minimax point estimation rate of theta does not deteriorate when $\varepsilon$ becomes unknown, our results reveal that, with a given $σ^2$, the minimax-optimal length of confidence intervals that are adaptive to unknown $\varepsilon$ is of order $σ(n^{-1/4}+\varepsilon^{1/2}/\max\{1, \log(en \varepsilon^2)\}^{1/2})$, which is polynomially worse than the optimal length when $\varepsilon$ is known. When the variance $σ^2$ is also unknown, we show a further degradation: no adaptive robust confidence interval can be shorter than $Ω(σn^{-1/8})$. Algorithmically, we introduce a Fourier-based certification procedure built on Carathéodory's positive-semidefiniteness constraints. By scanning candidate points and accepting those whose residual characteristic function is certifiably consistent with a Gaussian location mixture, our algorithm attains the minimax lower bound in the known-variance setting and is computable in polynomial time.

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