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Distributional results for thresholding estimators in high-dimensional Gaussian regression models

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  • Pötscher, Benedikt M.
  • Schneider, Ulrike

Abstract

We study the distribution of hard-, soft-, and adaptive soft-thresholding estimators within a linear regression model where the number of parameters k can depend on sample size n and may diverge with n. In addition to the case of known error-variance, we define and study versions of the estimators when the error-variance is unknown. We derive the finite-sample distribution of each estimator and study its behavior in the large-sample limit, also investigating the effects of having to estimate the variance when the degrees of freedom n-k does not tend to infinity or tends to infinity very slowly. Our analysis encompasses both the case where the estimators are tuned to perform consistent model selection and the case where the estimators are tuned to perform conservative model selection. Furthermore, we discuss consistency, uniform consistency and derive the minimax rate under either type of tuning.

Suggested Citation

  • 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.
    • Handle: RePEc:pra:mprapa:31882
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    References listed on IDEAS

    as
    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. Leeb, Hannes & Pötscher, Benedikt M., 2003. "The Finite-Sample Distribution Of Post-Model-Selection Estimators And Uniform Versus Nonuniform Approximations," Econometric Theory, Cambridge University Press, vol. 19(1), pages 100-142, February.
    3. Pötscher, Benedikt M. & Schneider, Ulrike, 2008. "Confidence sets based on penalized maximum likelihood estimators," MPRA Paper 9062, University Library of Munich, Germany.
    Full references (including those not matched with items on IDEAS)

    Citations

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    Cited by:

    1. Leeb, Hannes & Pötscher, Benedikt M. & Ewald, Karl, 2014. "On various confidence intervals post-model-selection," MPRA Paper 52858, University Library of Munich, Germany.
    2. 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.
    3. 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.
    4. William Kengne, 2023. "On consistency for time series model selection," Statistical Inference for Stochastic Processes, Springer, vol. 26(2), pages 437-458, July.

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    More about this item

    Keywords

    Thresholding; Lasso; adaptive Lasso; penalized maximum likelihood; finite-sample distribution; asymptotic distribution; variance estimation; minimax rate; high-dimensional model; oracle property;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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