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Equality between two general ridge estimators and equivalence of their residual sums of squares

Author

Listed:
  • Hirai Mukasa

    (Kyushu University)

  • Koji Tsukuda

    (Kyushu University)

Abstract

General ridge estimators are typical linear estimators in a general linear model. The class of them includes some shrinkage estimators in addition to classical linear unbiased estimators such as the ordinary least squares estimator and the weighted least squares estimator. Also, they have some properties such as linear sufficiency and admissibility in the class of linear estimators. We derive necessary and sufficient conditions under which two general ridge estimators coincide. In particular, two noteworthy conditions are added to those from previous studies. The first condition is given as a seemingly column space relationship to the covariance matrix of the error term, and the second one is based on the biases of general ridge estimators. Another problem studied in this paper is to derive an equivalence condition such that equality between two residual sums of squares holds when general ridge estimators are considered. Additionally, we demonstrate some concrete examples for which the equivalence conditions hold.

Suggested Citation

  • Hirai Mukasa & Koji Tsukuda, 2025. "Equality between two general ridge estimators and equivalence of their residual sums of squares," Statistical Papers, Springer, vol. 66(1), pages 1-20, January.
    • Handle: RePEc:spr:stpapr:v:66:y:2025:i:1:d:10.1007_s00362-024-01644-z
    DOI: 10.1007/s00362-024-01644-z
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    References listed on IDEAS

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    1. Markiewicz, Augustyn, 1996. "Characterization of general ridge estimators," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 145-148, April.
    2. Bernhard Arnold & Peter Stahlecker, 2011. "An unexpected property of minimax estimation in the relative squared error approach to linear regression analysis," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 74(3), pages 397-407, November.
    3. Koji Tsukuda & Hiroshi Kurata, 2020. "Covariance structure associated with an equality between two general ridge estimators," Statistical Papers, Springer, vol. 61(3), pages 1069-1084, June.
    Full references (including those not matched with items on IDEAS)

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