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 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 σ2. 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 σ2, the minimax-optimal length of confidence intervals that are adaptive to unknown varepsilon is of order σ (n-1/4+varepsilon1/2/max1, log(en varepsilon2)1/2), which is polynomially worse than the optimal length when varepsilon is known. When the variance σ2 is also unknown, we show a further degradation: no adaptive confidence interval can be shorter than Omega(σ n-1/8). Algorithmically, we introduce a Fourier-based certification procedure built on Carath'eodory'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.