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Density Prediction and the Stein Phenomenon

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  • Malay Ghosh

    (University of Florida)

  • Tatsuya Kubokawa

    (University of Tokyo)

  • Gauri Sankar Datta

    (University of Georgia)

Abstract

The Stein phenomenon is a path-breaking discovery in mathematical statistics in the last century. A large number of researchers followed Stein’s footsteps and developed a wide variety of minimax shrinkage point estimators of a multivariate normal mean vector, each dominating the sample mean. More recently, the problem resurfaced, but this time with minimax shrinkage predictive density estimation, illustrating once again the Stein phenomenon In this review paper, we discuss parallel developments for normal and Poisson distributions under the Kullback-Leibler and more general divergence losses.

Suggested Citation

  • Malay Ghosh & Tatsuya Kubokawa & Gauri Sankar Datta, 2020. "Density Prediction and the Stein Phenomenon," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(2), pages 330-352, August.
    • Handle: RePEc:spr:sankha:v:82:y:2020:i:2:d:10.1007_s13171-019-00186-z
    DOI: 10.1007/s13171-019-00186-z
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    References listed on IDEAS

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    1. Komaki, Fumiyasu, 2015. "Simultaneous prediction for independent Poisson processes with different durations," Journal of Multivariate Analysis, Elsevier, vol. 141(C), pages 35-48.
    2. Tsukuma, Hisayuki & Kubokawa, Tatsuya, 2017. "Proper Bayes and minimax predictive densities related to estimation of a normal mean matrix," Journal of Multivariate Analysis, Elsevier, vol. 159(C), pages 138-150.
    3. Takeru Matsuda & Fumiyasu Komaki, 2015. "Singular value shrinkage priors for Bayesian prediction," Biometrika, Biometrika Trust, vol. 102(4), pages 843-854.
    4. José Manuel Corcuera & Federica Giummolè, 1999. "A Generalized Bayes Rule for Prediction," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(2), pages 265-279, June.
    5. A. Boisbunon & Y. Maruyama, 2014. "Inadmissibility of the best equivariant predictive density in the unknown variance case," Biometrika, Biometrika Trust, vol. 101(3), pages 733-740.
    6. Kengo Kato, 2009. "Improved prediction for a multivariate normal distribution with unknown mean and variance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(3), pages 531-542, September.
    7. Ghosh, Malay & Mergel, Victor & Datta, Gauri Sankar, 2008. "Estimation, prediction and the Stein phenomenon under divergence loss," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 1941-1961, October.
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