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Estimation under matrix quadratic loss and matrix superharmonicity
[Shrinkage estimation with a matrix loss function]

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  • T Matsuda
  • W E Strawderman

Abstract

SummaryWe investigate estimation of a normal mean matrix under the matrix quadratic loss. Improved estimation under the matrix quadratic loss implies improved estimation of any linear combination of the columns under the quadratic loss. First, an unbiased estimate of risk is derived and the Efron–Morris estimator is shown to be minimax. Next, a notion of matrix superharmonicity for matrix-variate functions is introduced and shown to have properties analogous to those of the usual superharmonic functions, which may be of independent interest. Then, it is shown that the generalized Bayes estimator with respect to a matrix superharmonic prior is minimax. We also provide a class of matrix superharmonic priors that includes the previously proposed generalization of Stein’s prior. Numerical results demonstrate that matrix superharmonic priors work well for low-rank matrices.

Suggested Citation

  • T Matsuda & W E Strawderman, 2022. "Estimation under matrix quadratic loss and matrix superharmonicity [Shrinkage estimation with a matrix loss function]," Biometrika, Biometrika Trust, vol. 109(2), pages 503-519.
    • Handle: RePEc:oup:biomet:v:109:y:2022:i:2:p:503-519.
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    File URL: http://hdl.handle.net/10.1093/biomet/asab025
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    References listed on IDEAS

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    1. Honda, Toshio, 1991. "Minimax estimators in the manova model for arbitrary quadratic loss and unknown covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 36(1), pages 113-120, January.
    2. Takeru Matsuda & Fumiyasu Komaki, 2015. "Singular value shrinkage priors for Bayesian prediction," Biometrika, Biometrika Trust, vol. 102(4), pages 843-854.
    3. Bilodeau, Martin & Kariya, Takeaki, 1989. "Minimax estimators in the normal MANOVA model," Journal of Multivariate Analysis, Elsevier, vol. 28(2), pages 260-270, February.
    4. Matsuda, Takeru & Komaki, Fumiyasu, 2019. "Empirical Bayes matrix completion," Computational Statistics & Data Analysis, Elsevier, vol. 137(C), pages 195-210.
    5. Arjun K. Gupta & Daya K. Nagar, 2000. "Matrix-variate beta distribution," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 24, pages 1-11, January.
    6. Xie, M. Y., 1993. "All Admissible Linear Estimates of the Mean Matrix," Journal of Multivariate Analysis, Elsevier, vol. 44(2), pages 220-226, February.
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