TIME-UNIFORM, NONPARAMETRIC, NONASYMPTOTIC CONFIDENCE SEQUENCES

• 成果类型: Article
• 署名作者: Howard, Steven R.; Ramdas, Aaditya; McAuliffe, Jon; Sekhon, Jasjeet
• 署名单位: University of California System; University of California Berkeley; Carnegie Mellon University; Carnegie Mellon University; Yale University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/20-AOS1991
• 发表日期: 2021
• 页码: 1055-1080
• 关键词: boundary crossing probabilities; iterated logarithm; sequential-analysis; renewal theory; moment bounds; inequalities; LAW; variance; DESIGN; tests

摘要

A confidence sequence is a sequence of confidence intervals that is uniformly valid over an unbounded time horizon. Our work develops confidence sequences whose widths go to zero, with nonasymptotic coverage guarantees under nonparametric conditions. We draw connections between the Cramer-Chernoff method for exponential concentration, the law of the iterated logarithm (LIL) and the sequential probability ratio test-our confidence sequences are time-uniform extensions of the first; provide tight, nonasymptotic characterizations of the second; and generalize the third to nonparametric settings, including sub-Gaussian and Bernstein conditions, self-normalized processes and matrix martingales. We illustrate the generality of our proof techniques by deriving an empirical-Bernstein bound growing at a LIL rate, as well as a novel upper LIL for the maximum eigenvalue of a sum of random matrices. Finally, we apply our methods to covariance matrix estimation and to estimation of sample average treatment effect under the Neyman-Rubin potential outcomes model.