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Confidence Sets Based on Thresholding Estimators in High-Dimensional Gaussian Regression Models

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  • Ulrike Schneider

Abstract

We study confidence intervals based on hard-thresholding, soft-thresholding, and adaptive soft-thresholding in a linear regression model where the number of regressors k may depend on and diverge with sample size n . In addition to the case of known error variance, we define and study versions of the estimators when the error variance is unknown. In the known-variance case, we provide an exact analysis of the coverage properties of such intervals in finite samples. We show that these intervals are always larger than the standard interval based on the least-squares estimator. Asymptotically, the intervals based on the thresholding estimators are larger even by an order of magnitude when the estimators are tuned to perform consistent variable selection. For the unknown-variance case, we provide nontrivial lower bounds and a small numerical study for the coverage probabilities in finite samples. We also conduct an asymptotic analysis where the results from the known-variance case can be shown to carry over asymptotically if the number of degrees of freedom n − k tends to infinity fast enough in relation to the thresholding parameter.

Suggested Citation

  • Ulrike Schneider, 2016. "Confidence Sets Based on Thresholding Estimators in High-Dimensional Gaussian Regression Models," Econometric Reviews, Taylor & Francis Journals, vol. 35(8-10), pages 1412-1455, December.
    • Handle: RePEc:taf:emetrv:v:35:y:2016:i:8-10:p:1412-1455
    DOI: 10.1080/07474938.2015.1092798
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    References listed on IDEAS

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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. Alexandre Belloni & Victor Chernozhukov, 2011. "High Dimensional Sparse Econometric Models: An Introduction," Papers 1106.5242, arXiv.org, revised Sep 2011.
    3. Pötscher, Benedikt M. & Schneider, Ulrike, 2007. "On the distribution of the adaptive LASSO estimator," MPRA Paper 6913, University Library of Munich, Germany.
    4. Pötscher, Benedikt M. & Schneider, Ulrike, 2008. "Confidence sets based on penalized maximum likelihood estimators," MPRA Paper 9062, University Library of Munich, Germany.
    5. Bai, Jushan & Ng, Serena, 2008. "Forecasting economic time series using targeted predictors," Journal of Econometrics, Elsevier, vol. 146(2), pages 304-317, October.
    6. 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.
    7. 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.
    8. Pötscher, Benedikt M. & Schneider, Ulrike, 2011. "Distributional results for thresholding estimators in high-dimensional Gaussian regression models," MPRA Paper 31882, University Library of Munich, Germany.
    9. Mehmet Caner & Hao Helen Zhang, 2014. "Adaptive Elastic Net for Generalized Methods of Moments," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(1), pages 30-47, January.
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