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Performance Limits For Estimators Of The Risk Or Distribution Of Shrinkage-Type Estimators, And Some General Lower Risk-Bound Results

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  • Leeb, Hannes
  • Pötscher, Benedikt M.

Abstract

We consider the problem of estimating measures of precision of shrinkage-type estimators such as their risk or distribution. The notion of shrinkage-type estimators here refers to estimators such as the James–Stein estimator and Lasso-type estimators, in addition to “thresholding†estimators such as, e.g., Hodges' so-called superefficient estimator. Although the precision measures of such estimators typically can be estimated consistently, we show that they cannot be estimated uniformly consistently (even locally). This follows as a corollary to (locally) uniform lower bounds on the performance of estimators of the precision measures that we obtain in the paper. These lower bounds are typically quite large (e.g., they approach ½ or 1 depending on the situation considered). The analysis is based on some general lower risk bounds and related general results on the (non)existence of uniformly consistent estimators also obtained in the paper.A preliminary draft of the results in Section 3 of this paper was written in 1999.

Suggested Citation

  • Leeb, Hannes & Pötscher, 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(1), pages 69-97, February.
  • Handle: RePEc:cup:etheor:v:22:y:2006:i:01:p:69-97_06
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    Cited by:

    1. 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.
    2. Audrino, Francesco & Camponovo, Lorenzo & Roth, Constantin, 2015. "Testing the lag structure of assets’ realized volatility dynamics," Economics Working Paper Series 1501, University of St. Gallen, School of Economics and Political Science.
    3. Christian T. Brownlees & Giampiero M. Gallo, 2010. "Comparison of Volatility Measures: a Risk Management Perspective," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 8(1), pages 29-56, Winter.
    4. 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.
    5. 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.
    6. Tae-Hwy Lee & Zhou Xi & Ru Zhang, 2013. "Testing for Neglected Nonlinearity Using Regularized Artificial Neural Networks," Working Papers 201422, University of California at Riverside, Department of Economics, revised Apr 2012.
    7. Ganggang Xu & Suojin Wang & Jianhua Z. Huang, 2014. "Focused information criterion and model averaging based on weighted composite quantile regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(2), pages 365-381, June.
    8. Leeb, Hannes & Pötscher, Benedikt M. & Kivaranovic, Danijel, 2018. "Comment on "Model Confidence Bounds for Variable Selection" by Yang Li, Yuetian Luo, Davide Ferrari, Xiaonan Hu, and Yichen Qin," MPRA Paper 90655, University Library of Munich, Germany.
    9. Andrews, Donald W.K. & Guggenberger, Patrik, 2009. "Incorrect asymptotic size of subsampling procedures based on post-consistent model selection estimators," Journal of Econometrics, Elsevier, vol. 152(1), pages 19-27, September.

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