Introducing the b-value: combining unbiased and biased estimators from a sensitivity analysis perspective

2026-02-18

AI Summary

This study addresses the challenge of robustly combining an unbiased but low-precision estimator with a biased but high-precision estimator when the magnitude of the bias is unknown. From a sensitivity analysis perspective, the authors construct a sequence of confidence intervals indexed by the assumed bias magnitude to assess the robustness of inferences to potential bias. The key innovation is the introduction of the “b-value”—the critical threshold of the maximum tolerable relative bias beyond which the combined estimator loses statistical significance. For three canonical classes of combined estimators, the paper derives bias-dependent confidence intervals and advocates a soft-thresholding–based reporting scheme for the b-value, which achieves optimal robustness under unknown bias and minimizes worst-case risk, thereby overcoming the limitations of traditional approaches that focus solely on point estimation.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

In empirical research, when we have multiple estimators for the same parameter of interest, a central question arises: how do we combine unbiased but less precise estimators with biased but more precise ones to improve the inference? Under this setting, the point estimation problem has attracted considerable attention. In this paper, we focus on a less studied inference question: how can we conduct valid statistical inference in such settings with unknown bias? We propose a strategy to combine unbiased and biased estimators from a sensitivity analysis perspective. We derive a sequence of confidence intervals indexed by the magnitude of the bias, which enable researchers to assess how conclusions vary with the bias levels. Importantly, we introduce the notion of the b-value, a critical value of the unknown maximum relative bias at which combining estimators does not yield a significant result. We apply this strategy to three canonical combined estimators: the precision-weighted estimator, the pretest estimator, and the soft-thresholding estimator. For each estimator, we characterize the sequence of confidence intervals and determine the bias threshold at which the conclusion changes. Based on the theory, we recommend reporting the b-value based on the soft-thresholding estimator and its associated confidence intervals, which are robust to unknown bias and achieve the lowest worst-case risk among the alternatives.

Problem

• b-value
• sensitivity analysis
• statistical inference
• biased estimator
• unbiased estimator

Innovation

• b-value
• sensitivity analysis
• biased estimator
• confidence interval
• soft-thresholding estimator