Open Access
April 2021 Time-uniform, nonparametric, nonasymptotic confidence sequences
Steven R. Howard, Aaditya Ramdas, Jon McAuliffe, Jasjeet Sekhon
Author Affiliations +
Ann. Statist. 49(2): 1055-1080 (April 2021). DOI: 10.1214/20-AOS1991


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 Cramér–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.


Download Citation

Steven R. Howard. Aaditya Ramdas. Jon McAuliffe. Jasjeet Sekhon. "Time-uniform, nonparametric, nonasymptotic confidence sequences." Ann. Statist. 49 (2) 1055 - 1080, April 2021.


Received: 1 February 2019; Revised: 1 May 2020; Published: April 2021
First available in Project Euclid: 2 April 2021

Digital Object Identifier: 10.1214/20-AOS1991

Primary: 62G05 , 62L12
Secondary: 60B20 , 60G42

Keywords: Confidence sequence , empirical-Bernstein bound , finite LIL bound , matrix concentration , potential outcomes , sequential probability ratio test

Rights: Copyright © 2021 Institute of Mathematical Statistics

Vol.49 • No. 2 • April 2021
Back to Top