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The One Standard Error Rule for Model Selection: Does It Work?

Author

Listed:
  • Yuchen Chen

    (Carlson School of Management, University of Minnesota, Minneapolis, MN 55455, USA)

  • Yuhong Yang

    (School of Statistics, University of Minnesota, Minneapolis, MN 55455, USA)

Abstract

Previous research provided a lot of discussion on the selection of regularization parameters when it comes to the application of regularization methods for high-dimensional regression. The popular “One Standard Error Rule” (1se rule) used with cross validation (CV) is to select the most parsimonious model whose prediction error is not much worse than the minimum CV error. This paper examines the validity of the 1se rule from a theoretical angle and also studies its estimation accuracy and performances in applications of regression estimation and variable selection, particularly for Lasso in a regression framework. Our theoretical result shows that when a regression procedure produces the regression estimator converging relatively fast to the true regression function, the standard error estimation formula in the 1se rule is justified asymptotically. The numerical results show the following: 1. the 1se rule in general does not necessarily provide a good estimation for the intended standard deviation of the cross validation error. The estimation bias can be 50–100% upwards or downwards in various situations; 2. the results tend to support that 1se rule usually outperforms the regular CV in sparse variable selection and alleviates the over-selection tendency of Lasso; 3. in regression estimation or prediction, the 1se rule often performs worse. In addition, comparisons are made over two real data sets: Boston Housing Prices (large sample size n , small/moderate number of variables p ) and Bardet–Biedl data (large p , small n ). Data guided simulations are done to provide insight on the relative performances of the 1se rule and the regular CV.

Suggested Citation

  • Yuchen Chen & Yuhong Yang, 2021. "The One Standard Error Rule for Model Selection: Does It Work?," Stats, MDPI, vol. 4(4), pages 1-25, November.
    • Handle: RePEc:gam:jstats:v:4:y:2021:i:4:p:51-892:d:673038
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    References listed on IDEAS

    as
    1. Jiahua Chen & Zehua Chen, 2008. "Extended Bayesian information criteria for model selection with large model spaces," Biometrika, Biometrika Trust, vol. 95(3), pages 759-771.
    2. Harrison, David Jr. & Rubinfeld, Daniel L., 1978. "Hedonic housing prices and the demand for clean air," Journal of Environmental Economics and Management, Elsevier, vol. 5(1), pages 81-102, March.
    3. Wenjing Yang & Yuhong Yang, 2017. "Toward an objective and reproducible model choice via variable selection deviation," Biometrics, The International Biometric Society, vol. 73(1), pages 20-30, March.
    4. Zhang, Yongli & Yang, Yuhong, 2015. "Cross-validation for selecting a model selection procedure," Journal of Econometrics, Elsevier, vol. 187(1), pages 95-112.
    5. Chenglong Ye & Yi Yang & Yuhong Yang, 2018. "Sparsity Oriented Importance Learning for High-Dimensional Linear Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(524), pages 1797-1812, October.
    6. Hansheng Wang & Runze Li & Chih-Ling Tsai, 2007. "Tuning parameter selectors for the smoothly clipped absolute deviation method," Biometrika, Biometrika Trust, vol. 94(3), pages 553-568.
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