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

• URL: http://arxiv.org/abs/2604.26992v1
• Date: Tue, 28 Apr 2026 22:40:54 GMT
• Title: Adaptive Robust Confidence Intervals in Efron's Gaussian Two-Groups Model
• Abstract summary: Motivated by noise-oblivious nature of adversaries, we study confidence intervals for the null location parameter $$ in Efron's Gaussian two-groups fraction model.<n>We characterize the minimax-optimal length among confidence intervals with a prescribed coverage level uniformly over the unknown proportion contamination and all noise-oblivious adversaries.
• Score: 28.636700996382558
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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