Time-uniform, nonparametric, nonasymptotic confidence sequences

TL;DR

This paper introduces nonparametric, nonasymptotic confidence sequences that are valid over infinite time horizons, with widths shrinking to zero, applicable to various statistical models including matrix martingales and treatment effect estimation.

Contribution

It develops a general framework for confidence sequences with nonasymptotic guarantees, extending classical methods to nonparametric and matrix settings, and demonstrates broad applicability.

Findings

Derived a LIL-rate empirical-Bernstein bound

Established a new upper LIL for maximum eigenvalue of matrix sums

Applied confidence sequences to covariance and treatment effect estimation

Abstract

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 Cram\'er-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.