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September 2013 Statistical significance in high-dimensional linear models
Peter Bühlmann
Bernoulli 19(4): 1212-1242 (September 2013). DOI: 10.3150/12-BEJSP11


We propose a method for constructing $p$-values for general hypotheses in a high-dimensional linear model. The hypotheses can be local for testing a single regression parameter or they may be more global involving several up to all parameters. Furthermore, when considering many hypotheses, we show how to adjust for multiple testing taking dependence among the $p$-values into account. Our technique is based on Ridge estimation with an additional correction term due to a substantial projection bias in high dimensions. We prove strong error control for our $p$-values and provide sufficient conditions for detection: for the former, we do not make any assumption on the size of the true underlying regression coefficients while regarding the latter, our procedure might not be optimal in terms of power. We demonstrate the method in simulated examples and a real data application.


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Peter Bühlmann. "Statistical significance in high-dimensional linear models." Bernoulli 19 (4) 1212 - 1242, September 2013.


Published: September 2013
First available in Project Euclid: 27 August 2013

zbMATH: 1273.62173
MathSciNet: MR3102549
Digital Object Identifier: 10.3150/12-BEJSP11

Keywords: Global testing , Lasso , multiple testing , Ridge regression , Variable selection , Westfall–Young permutation procedure

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability


Vol.19 • No. 4 • September 2013
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