On Shrinkage Estimators and “Effective Degrees of Freedom”

Lynn R. LaMotte (), Julia Volaufova () and Simo Puntanen ()
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Lynn R. LaMotte: Biostatistics Program, LSU Health-NO, School of Public Health
Julia Volaufova: Biostatistics Program, LSU Health-NO, School of Public Health
Simo Puntanen: Tampere University, Faculty of Information Technology and Communication Sciences

Chapter Chapter 14 in Recent Developments in Multivariate and Random Matrix Analysis, 2020, pp 245-254 from Springer

Abstract: Abstract Explicit expressions for the estimated mean y ~ k = X β ~ k = H k y $$\tilde {\mathbf {y}}_k = X\tilde {\boldsymbol {\beta }}_k = H_k\mathbf {y}$$ and effective degrees of freedomν k = tr(H k) by penalized least squares, with penalty k||Dβ||2, can be found readily when X ′X + D ′D is nonsingular. We establish them here in general under only the condition that X be a non-zero matrix, and we show that the monotonicity properties that are known when X ′X is nonsingular also hold in general, but that they are affected by estimability of Dβ. We establish the relation between these penalized least squares estimators and least squares under the restriction that Dβ = 0.

Date: 2020
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DOI: 10.1007/978-3-030-56773-6_14

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