  This paper investigates the theoretical underpinnings of two fundamental
statistical inference problems, the construction of confidence sets and
large-scale simultaneous hypothesis testing, in the presence of heavy-tailed
data. With heavy-tailed observation noise, finite sample properties of the
least squares-based methods, typified by the sample mean, are suboptimal both
theoretically and empirically. In this paper, we demonstrate that the adaptive
Huber regression, integrated with the multiplier bootstrap procedure, provides
a useful robust alternative to the method of least squares. Our theoretical and
empirical results reveal the effectiveness of the proposed method, and
highlight the importance of having inference methods that are robust to heavy
tailedness.
