This paper studies the construction of adaptive confidence intervals under Huber's contamination model when the contamination proportion is unknown. For the robust confidence interval of a Gaussian mean, we show that the optimal length of an adaptive interval must be exponentially wider than that of a non-adaptive one. An optimal construction is achieved through simultaneous uncertainty quantification of quantiles at all levels. The results are further extended beyond the Gaussian location model by addressing a general family of robust hypothesis testing. In contrast to adaptive robust estimation, our findings reveal that the optimal length of an adaptive robust confidence interval critically depends on the distribution's shape.

翻译：本文研究了在污染比例未知的Huber污染模型下自适应置信区间的构建问题。针对高斯均值的鲁棒置信区间，我们证明自适应区间的最优长度必须比非自适应区间指数级地更宽。通过同时量化所有分位数水平的不确定性，实现了最优构造。通过处理一般族的鲁棒假设检验问题，结果进一步推广到高斯位置模型之外。与自适应鲁棒估计形成对比的是，我们的研究结果表明自适应鲁棒置信区间的最优长度关键取决于分布的形状特征。