讲座题目：A Nonasymptotic  Theory of Robustness

主讲人：加州大学圣地亚哥分校周文心博士

讲座时间:2017年12月5日（星期二）2:00-3:00

讲座地点：浙江工商大学综合楼601会议室

主讲人简介：周文心，2013  年获得香港科技大学统计学博士学位(导师:邵启满教授)，随后分别在澳大利亚墨尔本大学数学与统计学院（导师：Aurore  Delaigle）和普林斯顿大学（导师：范剑青教授）进行博士后研究。主要研究方向为：asymptotic theory in probability and  statistics, large-scale statistical inference, nonparametric and robust  statistics. 近年来有数十篇文章在概率统计学顶级学术期刊 The Annals of Probability, The Annals of  Statistics、JASA、JMLR 、Biometrics等发表。

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.