University of California, San Diego (加州大学圣地亚哥分校)
题目: A Nonasymptotic Theory of Robustness
摘要: Massive data are often contaminated by outliers and heavy-tailed errors. In the presence of heavy-tailed data, finite sample properties of the least squares-based methods, typified by the sample mean, are suboptimal both theoretically and empirically. To address this challenge, we propose the adaptive Huber regression for robust estimation and inference. The key observation is that the robustification parameter should adapt to sample size, dimension and moments for optimal tradeoff between bias and robustness. For heavy-tailed data with bounded $(1+\delta)$-th moment for some $\delta>0$, we establish a sharp phase transition for robust estimation of regression parameters in both finite dimensional and high dimensional settings: when $\delta \geq 1$, the estimator achieves sub-Gaussian rate of convergence without sub-Gaussian assumptions, while only a slower rate is available in the regime $0<\delta <1$ and the transition is smooth and optimal. In addition, nonasymptotic Bahadur representation and Wilks’ expansion for finite sample inference are derived when higher moments exist. Based on these results, we make a further step on developing uncertainty quantification methodologies, including the construction of confidence sets and multiple testing. We demonstrate that the adaptive Huber regression, combined with the multiplier bootstrap procedure, provides a useful robust alternative to the method of least squares.
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