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Effective degrees of freedom: a flawed metaphor

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  • Lucas Janson
  • William Fithian
  • Trevor J. Hastie

Abstract

To most applied statisticians, a fitting procedure’s degrees of freedom is synonymous with its model complexity, or its capacity for overfitting to data. In particular, the degrees of freedom is often used to parameterize the bias-variance trade-off in model selection. We argue that, on the contrary, model complexity and degrees of freedom may correspond very poorly. We exhibit and theoretically explore various fitting procedures for which the degrees of freedom is not monotonic in the model complexity parameter and can exceed the total dimension of the ambient space even in very simple settings. We show that the degrees of freedom for any nonconvex projection method can be unbounded.

Suggested Citation

  • Lucas Janson & William Fithian & Trevor J. Hastie, 2015. "Effective degrees of freedom: a flawed metaphor," Biometrika, Biometrika Trust, vol. 102(2), pages 479-485.
    • Handle: RePEc:oup:biomet:v:102:y:2015:i:2:p:479-485.
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    File URL: http://hdl.handle.net/10.1093/biomet/asv019
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    Cited by:

    1. Andrew L. Hicks & Mark S. Handcock & Narayan Sastry & Anne R. Pebley, 2018. "Sequential Neighborhood Effects: The Effect of Long-Term Exposure to Concentrated Disadvantage on Children’s Reading and Math Test Scores," Demography, Springer;Population Association of America (PAA), vol. 55(1), pages 1-31, February.
    2. Lasanthi C. R. Pelawa Watagoda & David J. Olive, 2021. "Comparing six shrinkage estimators with large sample theory and asymptotically optimal prediction intervals," Statistical Papers, Springer, vol. 62(5), pages 2407-2431, October.
    3. Iacopo Bernetti & Veronica Alampi Sottini & Lorenzo Bambi & Elena Barbierato & Tommaso Borghini & Irene Capecchi & Claudio Saragosa, 2020. "Urban Niche Assessment: An Approach Integrating Social Media Analysis, Spatial Urban Indicators and Geo-Statistical Techniques," Sustainability, MDPI, vol. 12(10), pages 1-26, May.
    4. Daniel A. Griffith & Yongwan Chun, 2016. "Evaluating Eigenvector Spatial Filter Corrections for Omitted Georeferenced Variables," Econometrics, MDPI, vol. 4(2), pages 1-12, June.
    5. Daniel Durstewitz, 2017. "A state space approach for piecewise-linear recurrent neural networks for identifying computational dynamics from neural measurements," PLOS Computational Biology, Public Library of Science, vol. 13(6), pages 1-33, June.
    6. Philip T. Reiss & Lei Huang & Pei‐Shien Wu & Huaihou Chen & Stan Colcombe, 2017. "Pointwise influence matrices for functional‐response regression," Biometrics, The International Biometric Society, vol. 73(4), pages 1092-1101, December.

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