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Empirical Bayes deconvolution estimates

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  • Bradley Efron

Abstract

An unknown prior density $g(\theta )$ has yielded realizations $\Theta _1,\ldots ,\Theta _N$. They are unobservable, but each $\Theta _i$ produces an observable value $X_i$ according to a known probability mechanism, such as $X_i\sim {\rm Po}(\Theta _i)$. We wish to estimate $g(\theta )$ from the observed sample $X_1,\ldots ,X_N$. Traditional asymptotic calculations are discouraging, indicating very slow nonparametric rates of convergence. In this article we show that parametric exponential family modelling of $g(\theta )$ can give useful estimates in moderate-sized samples. We illustrate the approach with a variety of real and artificial examples. Covariate information can be incorporated into the deconvolution process, leading to a more detailed theory of generalized linear mixed models.

Suggested Citation

  • Bradley Efron, 2016. "Empirical Bayes deconvolution estimates," Biometrika, Biometrika Trust, vol. 103(1), pages 1-20.
    • Handle: RePEc:oup:biomet:v:103:y:2016:i:1:p:1-20.
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    File URL: http://hdl.handle.net/10.1093/biomet/asv068
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    Cited by:

    1. Patrick Kline & Evan K Rose & Christopher R Walters, 2022. "Systemic Discrimination Among Large U.S. Employers [“Teachers and Student Achievement in the Chicago Public High Schools,”]," The Quarterly Journal of Economics, Oxford University Press, vol. 137(4), pages 1963-2036.
    2. Jochmans, Koen & Weidner, Martin, 2024. "Inference On A Distribution From Noisy Draws," Econometric Theory, Cambridge University Press, vol. 40(1), pages 60-97, February.
    3. Manuel Arellano & Stéphane Bonhomme, 2023. "Recovering Latent Variables by Matching," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 118(541), pages 693-706, January.
    4. Patrick Kline & Evan K Rose & Christopher R Walters, 2023. "Systemic Discrimination Among Large U.S. Employers," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 137(4), pages 1963-2036.
    5. Zhang Qi & Xu Zheng & Lai Yutong, 2021. "An Empirical Bayes approach for the identification of long-range chromosomal interaction from Hi-C data," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 20(1), pages 1-15, February.
    6. Patrick Kline, 2023. "A Comment on: “Invidious Comparisons: Ranking and Selection as Compound Decisions” by Jiaying Gu and Roger Koenker," Econometrica, Econometric Society, vol. 91(1), pages 47-52, January.
    7. Raffaella Giacomini & Sokbae Lee & Silvia Sarpietro, 2023. "A Robust Method for Microforecasting and Estimation of Random Effects," Papers 2308.01596, arXiv.org.
    8. Roger Koenker, 2017. "Bayesian deconvolution: an R vinaigrette," CeMMAP working papers 38/17, Institute for Fiscal Studies.
    9. Mukhopadhyay, Subhadeep & Wang, Kaijun, 2023. "On The Problem of Relevance in Statistical Inference," Econometrics and Statistics, Elsevier, vol. 25(C), pages 93-109.
    10. Gribok, Andrei & Agarwal, Vivek & Yadav, Vaibhav, 2020. "Performance of empirical Bayes estimation techniques used in probabilistic risk assessment," Reliability Engineering and System Safety, Elsevier, vol. 201(C).
    11. Roger Koenker, 2017. "Bayesian deconvolution: an R vinaigrette," CeMMAP working papers CWP38/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Roger Koenker & Jiaying Gu, 2019. "Minimalist G-modelling: A comment on Efron," CeMMAP working papers CWP13/19, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    13. Jiaying Gu & Roger Koenker, 2020. "Invidious Comparisons: Ranking and Selection as Compound Decisions," Papers 2012.12550, arXiv.org, revised Sep 2021.
    14. J. R. Lockwood & Katherine E. Castellano & Benjamin R. Shear, 2018. "Flexible Bayesian Models for Inferences From Coarsened, Group-Level Achievement Data," Journal of Educational and Behavioral Statistics, , vol. 43(6), pages 663-692, December.

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