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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
  • P tscher, 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 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.

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Bibliographic Info

Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 22 (2006)
Issue (Month): 01 (February)
Pages: 69-97

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Handle: RePEc:cup:etheor:v:22:y:2006:i:01:p:69-97_06

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Cited by:
  1. 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.
  2. 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.
  3. Hannes Leeb & Benedikt M. Poetscher, 2005. "Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator," Cowles Foundation Discussion Papers 1500, Cowles Foundation for Research in Economics, Yale University, revised Apr 2007.
  4. 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.

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