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On the degrees of freedom in shrinkage estimation

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  • Kato, Kengo

Abstract

We study the degrees of freedom in shrinkage estimation of regression coefficients. Generalizing the idea of the Lasso, we consider the problem of estimating the coefficients by minimizing the sum of squares with the constraint that the coefficients belong to a closed convex set. Based on a differential geometric approach, we derive an unbiased estimator of the degrees of freedom for this estimation method, under a smoothness assumption on the boundary of the closed convex set. The result presented in this paper is applicable to estimation with a wide class of constraints. As an application, we obtain a Cp type criterion and AIC for selecting tuning parameters.

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  • Kato, Kengo, 2009. "On the degrees of freedom in shrinkage estimation," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1338-1352, August.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:7:p:1338-1352
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    References listed on IDEAS

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    1. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
    2. Bradley Efron, 2004. "The Estimation of Prediction Error: Covariance Penalties and Cross-Validation," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 619-632, January.
    3. Li, Youjuan & Liu, Yufeng & Zhu, Ji, 2007. "Quantile Regression in Reproducing Kernel Hilbert Spaces," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 255-268, March.
    4. Kuriki, Satoshi & Takemura, Akimichi, 2000. "Shrinkage Estimation towards a Closed Convex Set with a Smooth Boundary," Journal of Multivariate Analysis, Elsevier, vol. 75(1), pages 79-111, October.
    5. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    6. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
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    Cited by:

    1. Samuel Vaiter & Charles Deledalle & Jalal Fadili & Gabriel Peyré & Charles Dossal, 2017. "The degrees of freedom of partly smooth regularizers," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(4), pages 791-832, August.
    2. Guillaume Allaire Pouliot & Zhen Xie, 2022. "Degrees of Freedom and Information Criteria for the Synthetic Control Method," Papers 2207.02943, arXiv.org.
    3. Xiaoli Gao, 2018. "A flexible shrinkage operator for fussy grouped variable selection," Statistical Papers, Springer, vol. 59(3), pages 985-1008, September.
    4. James Younker, 2022. "Calculating Effective Degrees of Freedom for Forecast Combinations and Ensemble Models," Discussion Papers 2022-19, Bank of Canada.
    5. María José Lombardía & Esther López‐Vizcaíno & Cristina Rueda, 2017. "Mixed generalized Akaike information criterion for small area models," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 180(4), pages 1229-1252, October.
    6. María José Lombardía & Esther López-Vizcaíno & Cristina Rueda, 2021. "Selection model for domains across time: application to labour force survey by economic activities," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 228-254, March.
    7. Rueda, Cristina, 2013. "Degrees of freedom and model selection in semiparametric additive monotone regression," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 88-99.
    8. Hettihewa, Samanthala & Saha, Shrabani & Zhang, Hanxiong, 2018. "Does an aging population influence stock markets? Evidence from New Zealand," Economic Modelling, Elsevier, vol. 75(C), pages 142-158.
    9. Justin B. Post & Howard D. Bondell, 2013. "Factor Selection and Structural Identification in the Interaction ANOVA Model," Biometrics, The International Biometric Society, vol. 69(1), pages 70-79, March.
    10. Hirose, Kei & Tateishi, Shohei & Konishi, Sadanori, 2013. "Tuning parameter selection in sparse regression modeling," Computational Statistics & Data Analysis, Elsevier, vol. 59(C), pages 28-40.
    11. Minami, Kentaro, 2020. "Degrees of freedom in submodular regularization: A computational perspective of Stein’s unbiased risk estimate," Journal of Multivariate Analysis, Elsevier, vol. 175(C).
    12. Luigi Augugliaro & Angelo M. Mineo & Ernst C. Wit, 2013. "Differential geometric least angle regression: a differential geometric approach to sparse generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(3), pages 471-498, June.

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