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

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 $\theta$ 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 $\sigma^2$. 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 $\sigma^2$, the minimax-optimal length of confidence intervals that are adaptive to unknown $\varepsilon$ is of order $\sigma (n^{-1/4}+\varepsilon^{1/2}/\max\{1, \log(en \varepsilon^2)\}^{1/2})$, which is polynomially worse than the optimal length when $\varepsilon$ is known. When the variance $\sigma^2$ is also unknown, we show a further degradation: no adaptive confidence interval can be shorter than $\Omega(\sigma n^{-1/8})$. Algorithmically, we introduce a Fourier-based certification procedure built on Carath\'{e}odory'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.