Abstract : We study the construction of confidence intervals under Huber's contamination model. When the contamination proportion is unknown, we characterize the necessary adaptation cost of the problem. In particular, for Gaussian location model, the optimal length of an adaptive confidence interval is proved to be exponentially wider than that of a non-adaptive one. Results for general location models will be discussed. In addition, we also consider the same problem in a network setting for an Erdos-Renyi graph with node contamination. It will be shown that the hardness of the adaptive confidence interval construction is implied by the detection threshold between Erdos-Renyi model and stochastic block model.

Keywords : adaptivity; confidence intervals; Huber's contamination model; location model; minimax testing; max hypothesis testing