Time-uniform, nonparametric, nonasymptotic confidence sequences (1810.08240v9)

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.

• The paper introduces time-uniform, nonparametric, nonasymptotic confidence sequences valid over indefinite time horizons, suitable for dynamic sequential experiments.
• It develops a theoretical framework using advanced statistical theories and generalized proof techniques for constructing these sequences under various conditions like sub-Gaussian or Bernstein.
• This work has practical implications for adaptive sequential analysis in industries like online platforms and clinical trials, enabling valid inference with flexible, non-deterministic stopping rules.

Summary of "Time-uniform, Nonparametric, Nonasymptotic Confidence Sequences"

The paper "Time-uniform, nonparametric, nonasymptotic confidence sequences" by Howard et al. presents a comprehensive exploration of confidence sequences as a method for deriving confidence intervals in sequential experiments. Confidence sequences are defined as a series of confidence intervals that are valid over an indefinite time horizon and offer a uniformly controlled error rate. This contrasts with typical confidence intervals, which are valid only at a fixed sample size, requiring assumptions that do not hold in dynamically changing environments like sequential experiments.

Core Contributions

1. Nonparametric and Nonasymptotic Approach: The paper advocates for constructing confidence sequences that are independent of asymptotic behaviors or specific parametric distribution assumptions. Instead, it introduces nonparametric confidence sequences, suitable for any sample size.
2. Theoretical Framework: Connections are drawn between advanced statistical theories, including the Cramér-Chernoff method for exponential concentration inequalities, the law of the iterated logarithm (LIL), and the sequential probability ratio test. This theoretical framework supports the development of time-uniform confidence sequences.
3. Generalized Proof Techniques: Through innovative proof techniques, the paper derives confidence sequences for various settings, such as sub-Gaussian and Bernstein conditions, self-normalized processes, and matrix martingales. Highlighted are empirical-Bernstein bounds and upper LILs for matrices' maximum eigenvalues.
4. Applications in Sequential Inference: By applying these general principles, the paper addresses applications in covariance matrix estimation and estimating sample average treatment effects, broadening the scope of existing sequential analysis methods.

Implications and Future Directions

Practical Implications

• Adaptive Sequential Analysis: This work significantly impacts industries relying on sequential experiments, such as online platforms and clinical trials. The model allows for adaptive sampling and decision-making without sacrificing inferential validity.
• Enhanced Flexibility: By allowing non-deterministic sampling sizes and arbitrary stopping rules, experimenters can respond more nimbly to incoming data.

Theoretical Implications

• Novel Statistical Boundaries: The paper introduces new statistical principles like the "stitching method" for deriving curved boundaries, adapting traditional fixed-sample techniques to sequential settings.
• Potential for Exploration: These techniques present opportunities for further exploration in statistical efficiency and optimality in sequential experimentation contexts.

Future Directions

The paper opens avenues for research in extending these methods to continuous-time martingales and Banach space-valued processes. Given the complexity and utility of these new methods, future work may optimize the computational aspects of these confidence sequences, especially when deployed in large-scale industrial or clinical trial contexts.

This paper sets the stage for both theoretical advancements and practical enhancements in sequential data analysis. The thorough structure allows for in-depth exploration into complex statistical problems, providing a solid foundation for next-generation methodologies in adaptive inference and decision-making.