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On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding

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  • Pötscher, Benedikt M.
  • Leeb, Hannes

Abstract

We study the distributions of the LASSO, SCAD, and thresholding estimators, in finite samples and in the large-sample limit. The asymptotic distributions are derived for both the case where the estimators are tuned to perform consistent model selection and for the case where the estimators are tuned to perform conservative model selection. Our findings complement those of Knight and Fu [K. Knight, W. Fu, Asymptotics for lasso-type estimators, Annals of Statistics 28 (2000) 1356-1378] and Fan and Li [J. Fan, R. Li, Variable selection via non-concave penalized likelihood and its oracle properties, Journal of the American Statistical Association 96 (2001) 1348-1360]. We show that the distributions are typically highly non-normal regardless of how the estimator is tuned, and that this property persists in large samples. The uniform convergence rate of these estimators is also obtained, and is shown to be slower than n-1/2 in case the estimator is tuned to perform consistent model selection. An impossibility result regarding estimation of the estimators' distribution function is also provided.

Suggested Citation

  • Pötscher, Benedikt M. & Leeb, Hannes, 2009. "On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding," Journal of Multivariate Analysis, Elsevier, vol. 100(9), pages 2065-2082, October.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:9:p:2065-2082
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    1. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    2. Leeb, Hannes & P tscher, Benedikt M., 2008. "Can One Estimate The Unconditional Distribution Of Post-Model-Selection Estimators?," Econometric Theory, Cambridge University Press, vol. 24(02), pages 338-376, April.
    3. Leeb, Hannes & Potscher, Benedikt M., 2008. "Sparse estimators and the oracle property, or the return of Hodges' estimator," Journal of Econometrics, Elsevier, vol. 142(1), pages 201-211, January.
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    5. Leeb, Hannes & P tscher, Benedikt M., 2005. "Model Selection And Inference: Facts And Fiction," Econometric Theory, Cambridge University Press, vol. 21(01), pages 21-59, February.
    6. Leeb, Hannes & P tscher, Benedikt M., 2006. "Performance Limits For Estimators Of The Risk Or Distribution Of Shrinkage-Type Estimators, And Some General Lower Risk-Bound Results," Econometric Theory, Cambridge University Press, vol. 22(01), pages 69-97, February.
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    8. Kabaila, Paul, 1995. "The Effect of Model Selection on Confidence Regions and Prediction Regions," Econometric Theory, Cambridge University Press, vol. 11(03), pages 537-549, June.
    9. Pötscher, Benedikt M., 2006. "The Distribution of Model Averaging Estimators and an Impossibility Result Regarding Its Estimation," MPRA Paper 73, University Library of Munich, Germany, revised Jul 2006.
    10. Leeb, Hannes & P tscher, Benedikt M., 2003. "The Finite-Sample Distribution Of Post-Model-Selection Estimators And Uniform Versus Nonuniform Approximations," Econometric Theory, Cambridge University Press, vol. 19(01), pages 100-142, February.
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    Cited by:

    1. Lu, Xun & Su, Liangjun, 2016. "Shrinkage estimation of dynamic panel data models with interactive fixed effects," Journal of Econometrics, Elsevier, vol. 190(1), pages 148-175.
    2. Giurcanu, Mihai C., 2012. "Bootstrapping in non-regular smooth function models," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 78-93.
    3. Kascha, Christian & Trenkler, Carsten, 2011. "Bootstrapping the likelihood ratio cointegration test in error correction models with unknown lag order," Computational Statistics & Data Analysis, Elsevier, vol. 55(2), pages 1008-1017, February.
    4. Anders Bredahl Kock, 2013. "Oracle inequalities for high-dimensional panel data models," CREATES Research Papers 2013-20, Department of Economics and Business Economics, Aarhus University.
    5. McCloskey, Adam, 2017. "Bonferroni-based size-correction for nonstandard testing problems," Journal of Econometrics, Elsevier, vol. 200(1), pages 17-35.
    6. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney K. Newey, 2016. "Double machine learning for treatment and causal parameters," CeMMAP working papers CWP49/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Leeb, Hannes & Pötscher, Benedikt M. & Ewald, Karl, 2014. "On various confidence intervals post-model-selection," MPRA Paper 52858, University Library of Munich, Germany.
    8. Marcelo C. Medeiros & Eduardo F. Mendes, 2015. "l1-Regularization of High-Dimensional Time-Series Models with Flexible Innovations," Textos para discussão 636, Department of Economics PUC-Rio (Brazil).
    9. repec:taf:jnlbes:v:34:y:2016:i:4:p:606-619 is not listed on IDEAS
    10. Farrell, Max H., 2015. "Robust inference on average treatment effects with possibly more covariates than observations," Journal of Econometrics, Elsevier, vol. 189(1), pages 1-23.
    11. repec:spr:compst:v:32:y:2017:i:2:d:10.1007_s00180-016-0690-2 is not listed on IDEAS
    12. Laurin Charles & Boomsma Dorret & Lubke Gitta, 2016. "The use of vector bootstrapping to improve variable selection precision in Lasso models," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 15(4), pages 305-320, August.
    13. Kwon, Sunghoon & Lee, Sangin & Kim, Yongdai, 2015. "Moderately clipped LASSO," Computational Statistics & Data Analysis, Elsevier, vol. 92(C), pages 53-67.
    14. Pötscher, Benedikt M., 2007. "Confidence Sets Based on Sparse Estimators Are Necessarily Large," MPRA Paper 5677, University Library of Munich, Germany.
    15. Pötscher, Benedikt M. & Schneider, Ulrike, 2007. "On the distribution of the adaptive LASSO estimator," MPRA Paper 6913, University Library of Munich, Germany.
    16. Pötscher, Benedikt M. & Schneider, Ulrike, 2008. "Confidence sets based on penalized maximum likelihood estimators," MPRA Paper 9062, University Library of Munich, Germany.
    17. Alexandre Belloni & Victor Chernozhukov & Ying Wei, 2016. "Post-Selection Inference for Generalized Linear Models With Many Controls," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(4), pages 606-619, October.
    18. Latouche, Pierre & Mattei, Pierre-Alexandre & Bouveyron, Charles & Chiquet, Julien, 2016. "Combining a relaxed EM algorithm with Occam’s razor for Bayesian variable selection in high-dimensional regression," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 177-190.
    19. Anders Bredahl Kock, 2012. "On the Oracle Property of the Adaptive Lasso in Stationary and Nonstationary Autoregressions," CREATES Research Papers 2012-05, Department of Economics and Business Economics, Aarhus University.
    20. Andreas Groll & Gerhard Tutz, 2017. "Variable selection in discrete survival models including heterogeneity," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(2), pages 305-338, April.
    21. Max H. Farrell, 2013. "Robust Inference on Average Treatment Effects with Possibly More Covariates than Observations," Papers 1309.4686, arXiv.org, revised Feb 2018.

    More about this item

    Keywords

    primary; 62J07; 62J05; 62F11; 62F12; 62E15 Penalized maximum likelihood LASSO SCAD Thresholding Post-model-selection estimator Finite-sample distribution Asymptotic distribution Oracle property Estimation of distribution Uniform consistency;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables

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