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On the distribution of the adaptive LASSO estimator

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  • Pötscher, Benedikt M.
  • Schneider, Ulrike

Abstract

We study the distribution of the adaptive LASSO estimator (Zou (2006)) in finite samples as well as in the large-sample limit. The large-sample distributions are derived both for the case where the adaptive LASSO estimator is tuned to perform conservative model selection as well as for the case where tuning results in consistent model selection. We show that the finite-sample as well as the large-sample distributions are typically highly non-normal, regardless of the choice of the tuning parameter. The uniform convergence rate is also obtained, and is shown to be slower than n^{-1/2} in case the estimator is tuned to perform consistent model selection. In particular, these results question the statistical relevance of the `oracle' property of the adaptive LASSO estimator established in Zou 2006). Moreover, we also provide an impossibility result regarding the estimation of the distribution function of the adaptive LASSO estimator.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 6913.

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Date of creation: Dec 2007
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Handle: RePEc:pra:mprapa:6913

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Keywords: Penalized maximum likelihood; LASSO; adaptive LASSO; nonnegative garotte; finite-sample distribution; asymptotic distribution; oracle property; estimation of distribution; uniform consistency;

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References

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  1. Pötscher, Benedikt M., 2007. "Confidence Sets Based on Sparse Estimators Are Necessarily Large," MPRA Paper 5677, University Library of Munich, Germany.
  2. Leeb, Hannes & P tscher, Benedikt M., 2008. "Can One Estimate The Unconditional Distribution Of Post-Model-Selection Estimators?," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 24(02), pages 338-376, April.
  3. Pötscher, Benedikt M. & Leeb, Hannes, 2007. "On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding," MPRA Paper 5615, University Library of Munich, Germany.
  4. Wang, Hansheng & Leng, Chenlei, 2007. "Unified LASSO Estimation by Least Squares Approximation," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 102, pages 1039-1048, September.
  5. Wang, Hansheng & Li, Guodong & Jiang, Guohua, 2007. "Robust Regression Shrinkage and Consistent Variable Selection Through the LAD-Lasso," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 25, pages 347-355, July.
  6. Hansheng Wang & Runze Li & Chih-Ling Tsai, 2007. "Tuning parameter selectors for the smoothly clipped absolute deviation method," Biometrika, Biometrika Trust, Biometrika Trust, vol. 94(3), pages 553-568.
  7. Kabaila, Paul, 1995. "The Effect of Model Selection on Confidence Regions and Prediction Regions," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 11(03), pages 537-549, June.
  8. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, Biometrika Trust, vol. 94(1), pages 19-35.
  9. 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, Cambridge University Press, vol. 19(01), pages 100-142, February.
  10. Hao Helen Zhang & Wenbin Lu, 2007. "Adaptive Lasso for Cox's proportional hazards model," Biometrika, Biometrika Trust, Biometrika Trust, vol. 94(3), pages 691-703.
  11. Hannes Leeb & Benedikt M. Poetscher, 2005. "Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 1500, Cowles Foundation for Research in Economics, Yale University, revised Apr 2007.
  12. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
  13. Hansheng Wang & Guodong Li & Chih-Ling Tsai, 2007. "Regression coefficient and autoregressive order shrinkage and selection via the lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, Royal Statistical Society, vol. 69(1), pages 63-78.
  14. Jianqing Fan & Runze Li, 2004. "New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 99, pages 710-723, January.
  15. 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.
  16. Leeb, Hannes & P tscher, Benedikt M., 2005. "Model Selection And Inference: Facts And Fiction," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 21(01), pages 21-59, February.
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Cited by:
  1. Ulrike Schneider & Martin Wagner, 2009. "Catching Growth Determinants with the Adaptive Lasso," wiiw Working Papers, The Vienna Institute for International Economic Studies, wiiw 55, The Vienna Institute for International Economic Studies, wiiw.
  2. Pötscher, Benedikt M. & Schneider, Ulrike, 2008. "Confidence sets based on penalized maximum likelihood estimators," MPRA Paper 9062, University Library of Munich, Germany.

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