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Weak Versus Strong Dominance of Shrinkage Estimators

Author

Listed:
  • Giuseppe Luca

    (University of Palermo)

  • Jan R. Magnus

    (Vrije Universiteit Amsterdam
    Tinbergen Institute)

Abstract

We consider the estimation of the mean of a multivariate normal distribution with known variance. Most studies consider the risk of competing estimators, that is the trace of the mean squared error matrix. In contrast we consider the whole mean squared error matrix, in particular its eigenvalues. We prove that there are only two distinct eigenvalues and apply our findings to the James–Stein and the Thompson class of estimators. It turns out that the famous Stein paradox is no longer a paradox when we consider the whole mean squared error matrix rather than only its trace.

Suggested Citation

  • Giuseppe Luca & Jan R. Magnus, 2021. "Weak Versus Strong Dominance of Shrinkage Estimators," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 239-266, December.
    • Handle: RePEc:spr:jqecon:v:19:y:2021:i:1:d:10.1007_s40953-021-00270-y
    DOI: 10.1007/s40953-021-00270-y
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    References listed on IDEAS

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    1. Magnus, Jan R., 1982. "Multivariate error components analysis of linear and nonlinear regression models by maximum likelihood," Journal of Econometrics, Elsevier, vol. 19(2-3), pages 239-285, August.
    2. Bruce E. Hansen, 2016. "The Risk of James--Stein and Lasso Shrinkage," Econometric Reviews, Taylor & Francis Journals, vol. 35(8-10), pages 1456-1470, December.
    3. Jan R. Magnus & Giuseppe De Luca, 2016. "Weighted-Average Least Squares (Wals): A Survey," Journal of Economic Surveys, Wiley Blackwell, vol. 30(1), pages 117-148, February.
    4. Hansen, Bruce E., 2015. "Shrinkage Efficiency Bounds," Econometric Theory, Cambridge University Press, vol. 31(4), pages 860-879, August.
    5. Casella, George, 1990. "Estimators with nondecreasing risk: application of a chi-squared identity," Statistics & Probability Letters, Elsevier, vol. 10(2), pages 107-109, July.
    6. Jan R. Magnus, 2002. "Estimation of the mean of a univariate normal distribution with known variance," Econometrics Journal, Royal Economic Society, vol. 5(1), pages 225-236, June.
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    Cited by:

    1. Yong Bao & Aman Ullah, 2021. "The Special Issue in Honor of Anirudh Lal Nagar: An Introduction," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 1-8, December.

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    More about this item

    Keywords

    Shrinkage; Dominance; James–Stein;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation

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