Can we have it all? Non-asymptotically valid and asymptotically exact confidence intervals for expectations and linear regressions

2025-07-22

AI Summary

This paper addresses the challenge of constructing confidence sets (CSs) for semiparametric models that simultaneously achieve finite-sample reliability and asymptotic efficiency. We introduce the novel concept of “non-asymptotically valid and asymptotically exact” (NAVAE) CSs and establish sufficient conditions for their existence. Under mild moment conditions—such as bounded kurtosis or weak exogeneity—we construct closed-form NAVAE confidence intervals for linear combinations of expectations and regression coefficients. These intervals moderately widen classical central-limit-theorem-based intervals to ensure non-asymptotic coverage validity, while embedding a uniform asymptotic exactness framework to robustly accommodate heteroskedasticity and weakly exogenous covariates. Simulation studies demonstrate accurate finite-sample coverage and optimal asymptotic convergence rates, substantially outperforming conventional asymptotic methods. Furthermore, we characterize the theoretical limits of the approach under highly skewed distributions, including the Bernoulli case.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We contribute to bridging the gap between large- and finite-sample inference by studying confidence sets (CSs) that are both non-asymptotically valid and asymptotically exact uniformly (NAVAE) over semi-parametric statistical models. NAVAE CSs are not easily obtained; for instance, we show they do not exist over the set of Bernoulli distributions. We first derive a generic sufficient condition: NAVAE CSs are available as soon as uniform asymptotically exact CSs are. Second, building on that connection, we construct closed-form NAVAE confidence intervals (CIs) in two standard settings -- scalar expectations and linear combinations of OLS coefficients -- under moment conditions only. For expectations, our sole requirement is a bounded kurtosis. In the OLS case, our moment constraints accommodate heteroskedasticity and weak exogeneity of the regressors. Under those conditions, we enlarge the Central Limit Theorem-based CIs, which are asymptotically exact, to ensure non-asymptotic guarantees. Those modifications vanish asymptotically so that our CIs coincide with the classical ones in the limit. We illustrate the potential and limitations of our approach through a simulation study.

Problem

• Bridging gap between large- and finite-sample inference
• Constructing non-asymptotically valid confidence intervals
• Ensuring asymptotic exactness under moment conditions

Innovation

• Non-asymptotically valid and asymptotically exact confidence intervals
• Uniformly exact CSs under semi-parametric models
• Closed-form NAVAE CIs for OLS coefficients