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Sparse Estimators and the Oracle Property, or the Return of Hodges' Estimator

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Author Info
Hannes Leeb () (Dept. of Statistics, Yale University)
Benedikt M. Poetscher (Dept. of Statistics, University of Vienna)

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Abstract

We point out some pitfalls related to the concept of an oracle property as used in Fan and Li (2001, 2002, 2004) which are reminiscent of the well-known pitfalls related to Hodges’ estimator. The oracle property is often a consequence of sparsity of an estimator. We show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function; in particular, the maximal risk diverges to infinity when ever the loss function is unbounded. For ease of presentation the result is set in the framework of a linear regression model, but generalizes far beyond that setting. In a Monte Carlo study we also assess the extent of the problem infinite samples for the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li (2001). We find that this estimator can perform rather poorly infinite samples and that its worst-case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity.

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File URL: http://cowles.econ.yale.edu/P/cd/d15a/d1500.pdf
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Publisher Info
Paper provided by Cowles Foundation, Yale University in its series Cowles Foundation Discussion Papers with number 1500.

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Length: 18 pages
Date of creation: Feb 2005
Date of revision: Apr 2007
Handle: RePEc:cwl:cwldpp:1500

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Related research
Keywords: Oracle property; Sparsity; Penalized maximum likelihood; Penalized least squares; Hodges’ estimator; SCAD; Lasso; Bridge estimator; Hard-thresholding; Maximal risk; Maximal absolute bias; Non-uniform limits;

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Find related papers by JEL classification:
C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General
C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation

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References listed on IDEAS
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  1. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December. [Downloadable!] (restricted)
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Pötscher, Benedikt M. & Schneider, Ulrike, 2007. "On the distribution of the adaptive LASSO estimator," MPRA Paper 6913, University Library of Munich, Germany, revised Dec 2008. [Downloadable!]
  2. 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, revised Mar 2009. [Downloadable!]
    Other versions:
  3. Pötscher, Benedikt M. & Schneider, Ulrike, 2008. "Confidence sets based on penalized maximum likelihood estimators," MPRA Paper 9062, University Library of Munich, Germany, revised May 2009. [Downloadable!]
  4. Jörg Stoye, 2008. "More on confidence intervals for partially identified parameters," CeMMAP working papers CWP11/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies. [Downloadable!]
    Other versions:
  5. Pötscher, Benedikt M., 2007. "Confidence Sets Based on Sparse Estimators Are Necessarily Large," MPRA Paper 5677, University Library of Munich, Germany, revised Apr 2009. [Downloadable!]
  6. Schneider, Ulrike & Wagner, Martin, 2008. "Catching Growth Determinants with the Adaptive LASSO," Economics Series 232, Institute for Advanced Studies. [Downloadable!]
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