Adaptive Robust Confidence Intervals

TL;DR

This paper develops adaptive confidence intervals under contamination models, revealing that their optimal length depends on the distribution shape and can be exponentially wider than non-adaptive intervals, with a construction based on simultaneous quantile uncertainty.

Contribution

It introduces a method for constructing optimal adaptive robust confidence intervals that account for unknown contamination levels and distribution shapes.

Findings

Adaptive intervals are exponentially wider than non-adaptive ones.

Optimal construction uses simultaneous quantile uncertainty quantification.

Distribution shape critically affects the interval length.

Abstract

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.