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Detection Boundary in Sparse Regression

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  • Yu I. Ingster

    (Crest)

  • Alexandre B. Tsybakov

    (Crest)

  • N. Verzelzn

    (Crest)

Abstract

We study the problem of detection of a p-dimensional sparse vector ofparameters in the linear regression model with Gaussian noise. We establishthe detection boundary, i.e., the necessary and sufficient conditions for thepossibility of successful detection as both the sample size n and the dimensionp tend to the infinity. Testing procedures that achieve this boundary arealso exhibited. Our results encompass the high-dimensional setting (p » n).The main message is that, under some conditions, the detection boundaryphenomenon that has been proved for the Gaussian sequence model, extendsto high-dimensional linear regression. Finally, we establish the detectionboundaries when the variance of the noise is unknown. Interestingly, thedetection boundaries sometimes depend on the knowledge of the variance ina high-dimensional setting.

Suggested Citation

  • Yu I. Ingster & Alexandre B. Tsybakov & N. Verzelzn, 2010. "Detection Boundary in Sparse Regression," Working Papers 2010-28, Center for Research in Economics and Statistics.
    • Handle: RePEc:crs:wpaper:2010-28
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    Cited by:

    1. Ian W. McKeague & Min Qian, 2015. "An Adaptive Resampling Test for Detecting the Presence of Significant Predictors," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(512), pages 1422-1433, December.
    2. Sardy, Sylvain & Diaz-Rodriguez, Jairo & Giacobino, Caroline, 2022. "Thresholding tests based on affine LASSO to achieve non-asymptotic nominal level and high power under sparse and dense alternatives in high dimension," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    3. Ian Barnett & Rajarshi Mukherjee & Xihong Lin, 2017. "The Generalized Higher Criticism for Testing SNP-Set Effects in Genetic Association Studies," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 64-76, January.
    4. He, Yi & Jaidee, Sombut & Gao, Jiti, 2023. "Most powerful test against a sequence of high dimensional local alternatives," Journal of Econometrics, Elsevier, vol. 234(1), pages 151-177.
    5. Rui Wang & Xingzhong Xu, 2021. "A Bayesian-motivated test for high-dimensional linear regression models with fixed design matrix," Statistical Papers, Springer, vol. 62(4), pages 1821-1852, August.
    6. Ery Arias-Castro & Meng Wang, 2017. "Distribution-free tests for sparse heterogeneous mixtures," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(1), pages 71-94, March.
    7. Matthias Löffler & Richard Nickl, 2017. "Comments on: High-dimensional simultaneous inference with the bootstrap," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(4), pages 731-733, December.

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