Zhexiao Lin, Peter J. Bickel, Peng Ding
arXiv 18 Feb 2026 · Statistics — Methodology
arXiv:2602.16310 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Le Cam, Lucien (1956) On the asymptotic theory of estimation and testing hypotheses | 1.000 | 6 | 4 | 100% |
| 2 | Bickel, PJ (1984) Parametric robustness: small biases can be worthwhile self | 0.737 | 10 | 4 | 40% |
| 3 | Green, Edwin J and Strawderman, William E (1991) A James-Stein type estimator for combining unbiased and possibly biased estimators | 0.737 | 3 | 3 | 67% |
| 4 | Angrist, Joshua D and Krueger, Alan B (1991) Does Compulsory School Attendance Affect Schooling and Earnings? | 0.737 | 3 | 2 | 100% |
| 5 | Armstrong, Timothy B and Kline, Patrick and Sun, Liyang (2025) Adapting to misspecification | 0.737 | 3 | 2 | 100% |
| 6 | J.[O]. Berger (1981) Estimation in continuous exponential families: Bayesian estimation subject to risk restrictions and inadmissibility results | 0.669 | 10 | 3 | 30% |
| 7 | Bickel, PJ (1983) Minimax estimation of the mean of a normal distribution subject to doing well at a point self | 0.644 | 2 | 2 | 100% |
| 8 | Giles, Judith A and Giles, David EA (1993) Pre-test estimation and testing in econometrics: recent developments | 0.644 | 2 | 2 | 100% |
| 9 | Rosenman, Evan TR and Basse, Guillaume and Owen, Art B and Baiocchi,… (2023) Combining observational and experimental datasets using shrinkage estimators | 0.644 | 2 | 2 | 100% |
| 10 | VanderWeele, Tyler J and Ding, Peng (2017) Sensitivity Analysis in Observational Research: Introducing the E-Value self | 0.644 | 2 | 2 | 100% |
Showing the top 10 of 42 scored citations.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Approximate Minimax Estimation of a Bounded Normal Mean via Stochastic Mirror Ascent | 0.405 | 1 | 1 |