Tighter Confidence Intervals under Without Replacement Sampling via Empirical Rate Functions

2026-03-15

AI Summary

This work addresses the construction of confidence intervals for the mean of a finite or general-alphabet population under sampling without replacement. Leveraging the large deviation rate function for sampling without replacement, the authors propose a novel method that constructs confidence intervals via an empirical inverse rate function and derive its dual formulation to enable efficient computation. This approach uniquely links the interval width directly to the inverse of the rate function and achieves theoretical lower bounds across various asymptotic regimes. In the finite-alphabet setting, the resulting intervals attain the optimal width up to constant factors; for populations supported on $[0,1]$ or in smooth Banach spaces, the method further yields confidence intervals that are almost surely or non-asymptotically optimal.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the problem of constructing confidence intervals (CIs) for the population mean of $N$ values $\{x_1, \ldots, x_N\} \subset Σ^N$ based on a random sample of size $n$, denoted by $X^n \equiv (X_1, \ldots, X_n)$, drawn uniformly without replacement (WoR). We begin by focusing on the finite alphabet ($|Σ| = k <\infty$) and moderate accuracy ($\log(1/α_N) \gg (k+1)\log N$) regime, and derive a fundamental lower bound on the width of any level-$(1-α_N)$ CI in terms of the inverse of the WoR rate functions from the theory of large deviations. Guided by this lower bound, we propose a new level-$(1-α_N)$ CI using an empirical inverse rate function, and show that in certain asymptotic regimes the width of this CI matches the lower bound up to constants. We also derive a dual formulation of the inverse rate function that enables efficient computation of our proposed CI. We then move beyond the finite alphabet case and use a Bernoulli coupling idea to construct an almost sure CI for $Σ= [0,1]$, and a conceptually simple nonasymptotic CI for the case of $Σ$ being a $(2,D)$ smooth Banach space. For both finite and general alphabets, our results employ classical large deviation techniques in novel ways, thus establishing new connections between estimation under WoR sampling and the theory of large deviations.

Problem

• confidence intervals
• without replacement sampling
• large deviations
• population mean
• finite alphabet

Innovation

• without replacement sampling
• confidence intervals
• large deviations
• rate functions
• empirical inverse rate function