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A Survey of Nonparametric Mixing Density Estimation via the Predictive Recursion Algorithm

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  • Ryan Martin

    (North Carolina State University)

Abstract

Nonparametric estimation of a mixing density based on observations from the corresponding mixture is a challenging statistical problem. This paper surveys the literature on a fast, recursive estimator based on the predictive recursion algorithm. After introducing the algorithm and giving a few examples, I summarize the available asymptotic convergence theory, describe an important semiparametric extension, and highlight two interesting applications. I conclude with a discussion of several recent developments in this area and some open problems.

Suggested Citation

  • Ryan Martin, 2021. "A Survey of Nonparametric Mixing Density Estimation via the Predictive Recursion Algorithm," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 97-121, May.
    • Handle: RePEc:spr:sankhb:v:83:y:2021:i:1:d:10.1007_s13571-019-00206-w
    DOI: 10.1007/s13571-019-00206-w
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    References listed on IDEAS

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    1. Wesley Tansey & Oluwasanmi Koyejo & Russell A. Poldrack & James G. Scott, 2018. "False Discovery Rate Smoothing," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(523), pages 1156-1171, July.
    2. Huageng Tao & Mari Palta & Brian S. Yandell & Michael A. Newton, 1999. "An Estimation Method for the Semiparametric Mixed Effects Model," Biometrics, The International Biometric Society, vol. 55(1), pages 102-110, March.
    3. Ryan Martin & Surya T. Tokdar, 2011. "Semiparametric inference in mixture models with predictive recursion marginal likelihood," Biometrika, Biometrika Trust, vol. 98(3), pages 567-582.
    4. P. Richard Hahn & Ryan Martin & Stephen G. Walker, 2018. "On Recursive Bayesian Predictive Distributions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(523), pages 1085-1093, July.
    5. Sylvia. Richardson & Peter J. Green, 1997. "On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(4), pages 731-792.
    6. Yong Wang, 2007. "On fast computation of the non‐parametric maximum likelihood estimate of a mixing distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(2), pages 185-198, April.
    7. Martin, Ryan & Han, Zhen, 2016. "A semiparametric scale-mixture regression model and predictive recursion maximum likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 75-85.
    8. Efron, Bradley, 2004. "Large-Scale Simultaneous Hypothesis Testing: The Choice of a Null Hypothesis," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 96-104, January.
    9. Ghosal,Subhashis & van der Vaart,Aad, 2017. "Fundamentals of Nonparametric Bayesian Inference," Cambridge Books, Cambridge University Press, number 9780521878265.
    10. Chae, Minwoo & Martin, Ryan & Walker, Stephen G., 2018. "Convergence of an iterative algorithm to the nonparametric MLE of a mixing distribution," Statistics & Probability Letters, Elsevier, vol. 140(C), pages 142-146.
    11. Martin, Ryan, 2012. "Convergence rate for predictive recursion estimation of finite mixtures," Statistics & Probability Letters, Elsevier, vol. 82(2), pages 378-384.
    12. James G. Scott & Ryan C. Kelly & Matthew A. Smith & Pengcheng Zhou & Robert E. Kass, 2015. "False Discovery Rate Regression: An Application to Neural Synchrony Detection in Primary Visual Cortex," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 459-471, June.
    13. Jin, Jiashun & Cai, T. Tony, 2007. "Estimating the Null and the Proportion of Nonnull Effects in Large-Scale Multiple Comparisons," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 495-506, June.
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