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Group bound: confidence intervals for groups of variables in sparse high dimensional regression without assumptions on the design

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  • Nicolai Meinshausen

Abstract

type="main" xml:id="rssb12094-abs-0001"> It is in general challenging to provide confidence intervals for individual variables in high dimensional regression without making strict or unverifiable assumptions on the design matrix. We show here that a ‘group bound’ confidence interval can be derived without making any assumptions on the design matrix. The lower bound for the regression coefficient of individual variables can be derived via linear programming. The idea also generalizes naturally to groups of variables, where we can derive a one-sided confidence interval for the joint effect of a group. Although the confidence intervals of individual variables are by the nature of the problem often very wide, it is shown to be possible to detect the contribution of groups of highly correlated predictor variables even when no variable individually shows a significant effect. The assumptions that are necessary to detect the effect of groups of variables are shown to be weaker than the weakest known assumptions that are necessary to detect the effect of individual variables.

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  • Nicolai Meinshausen, 2015. "Group bound: confidence intervals for groups of variables in sparse high dimensional regression without assumptions on the design," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(5), pages 923-945, November.
    • Handle: RePEc:bla:jorssb:v:77:y:2015:i:5:p:923-945
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    File URL: http://hdl.handle.net/10.1111/rssb.2015.77.issue-5
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    Cited by:

    1. Qing Zhou & Seunghyun Min, 2017. "Uncertainty quantification under group sparsity," Biometrika, Biometrika Trust, vol. 104(3), pages 613-632.
    2. Sauvenier, Mathieu & Van Bellegem, Sébastien, 2023. "Goodness-of-fit test in high-dimensional linear sparse models," LIDAM Discussion Papers CORE 2023008, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Ruben Dezeure & Peter Bühlmann & Cun-Hui Zhang, 2017. "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 685-719, December.
    4. Byol Kim & Song Liu & Mladen Kolar, 2021. "Two‐sample inference for high‐dimensional Markov networks," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(5), pages 939-962, November.

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