Anytime-Valid Sequential t-Tests and Confidence Sequences for Gaussian Means with Unknown Variance

Keskeiset käsitteet

This paper develops new e-processes and confidence sequences for sequential t-tests on the mean of a Gaussian distribution with unknown variance. The methods provide anytime-valid, nonasymptotic error control and optimal asymptotic performance.

Tiivistelmä

The paper focuses on constructing efficient and powerful sequential tests and confidence sequences for the mean of a Gaussian distribution with unknown variance.
Key highlights:

The authors apply the method of "Universal Inference" to derive e-processes and confidence sequences for both the point null (μ = μ0) and one-sided composite null (μ ≤ μ0) hypotheses. These provide anytime-valid error control and optimal asymptotic performance.

The authors revisit Lai's [1976] seminal work on sequential t-tests, filling in missing details and showing why his approach yields a confidence sequence without an e-process. They then introduce a simple variant that provides closed-form e-processes with improved tightness.

The authors derive information-theoretic lower bounds on the width of confidence intervals and upper bounds on the growth of e-processes for the t-test problem. They show that their new methods (from contributions 1 and 3) attain or nearly attain these optimal rates, while other existing conservative t-tests, and even the classical fixed-sample t-test, fail to be optimal.

The paper provides a comprehensive treatment of sequential t-tests, connecting them to group-invariant tests, likelihood ratio martingales, and universal inference, leading to a deeper understanding of the problem.

Tilastot

The sample mean b_μn and sample standard deviation √v_n are used as point estimators for μ and σ.
The t-statistic T_n-1 = √(n-1) (S_n - nμ_0) / √(nV_n - S_n^2) is a key quantity used in the analysis.

Lainaukset

"The classical duality between tests and interval estimates also manifests in the sequential setting."
"Nonnegative supermartingales are the prototypical examples of e-processes."
"The growth rate of a confidence sequence, i.e., its dependence on α → 0, is in the time-dependent form of α^(-1/n) which is worse than the typical known-variance rate √log(1/α)."

Tärkeimmät oivallukset

Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance

by Hongjian Wan... klo arxiv.org 04-25-2024

https://arxiv.org/pdf/2310.03722.pdf

Syvällisempiä Kysymyksiä

How can the methods developed in this paper be extended to more general settings beyond the Gaussian mean with unknown variance, such as regression models or non-i.i.d. data

The methods developed in the paper for Gaussian mean with unknown variance can be extended to more general settings by considering regression models or non-i.i.d. data.
For regression models, the concept of scale invariance can be applied to the predictors and response variables. By defining scale-invariant functions of the predictors and response, similar to the reduction of observations in the Gaussian setting, one can construct likelihood ratio martingales or e-processes for regression coefficients. This approach would allow for sequential testing and confidence intervals in regression analysis, ensuring time-uniform control of type 1 errors.
When dealing with non-i.i.d. data, the scale-invariant filtration can be adapted to capture the relative sizes or magnitudes of the observations. By considering scale-invariant reductions of the data, one can construct likelihood ratio martingales or e-processes that account for the non-i.i.d. nature of the data. This extension would enable sequential testing and confidence sequences in scenarios where the data does not follow an i.i.d. distribution.
Overall, by incorporating the principles of scale invariance and universal inference into more general settings, such as regression models and non-i.i.d. data, the methods developed in the paper can be extended to provide sequential t-tests and confidence sequences in a broader range of statistical applications.

What are the implications of the surprising α-suboptimality of the classical fixed-sample t-test

The surprising α-suboptimality of the classical fixed-sample t-test has important implications for sequential testing. The classical t-test, while widely used and well-established, may not always be the most efficient or optimal choice in sequential settings. The α-suboptimality suggests that the classical t-test may not achieve the best balance between error control and statistical power when applied sequentially.
To improve the performance of the classical t-test in the sequential setting, one approach could be to incorporate the insights from the scale-invariant approach and the universal inference approach. By combining these methodologies, one can potentially develop more powerful sequential t-tests that offer better control of type 1 errors at arbitrary stopping times while maximizing statistical power.
Additionally, exploring alternative test statistics, such as those derived from likelihood ratio martingales or e-processes, could lead to sequential testing procedures that outperform the classical fixed-sample t-test in terms of efficiency, optimality, and robustness in various statistical scenarios.

Are there ways to improve its performance in the sequential setting

The insights from the scale-invariant approach and the universal inference approach can be combined to yield more powerful sequential t-tests and confidence sequences. By integrating the principles of scale invariance with the methodology of universal inference, one can create a comprehensive framework for sequential hypothesis testing and interval estimation.
In this combined approach, scale invariance can be used to ensure that the statistical procedures are robust to changes in scale or magnitude of the data, while universal inference can provide a principled way to construct e-processes and likelihood ratio martingales for sequential testing.
By leveraging the strengths of both approaches, researchers can develop sequential t-tests and confidence sequences that offer time-uniform control of type 1 errors, optimal statistical power, and adaptability to various data distributions and settings. This integration of scale invariance and universal inference can lead to more efficient and effective sequential statistical methods for a wide range of applications.