May 2021 Optimal sparsity testing in linear regression model
Alexandra Carpentier, Nicolas Verzelen
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Bernoulli 27(2): 727-750 (May 2021). DOI: 10.3150/20-BEJ1224


We consider the problem of sparsity testing in the high-dimensional linear regression model. The problem is to test whether the number of non-zero components (aka the sparsity) of the regression parameter θ is less than or equal to k0. We pinpoint the minimax separation distances for this problem, which amounts to quantifying how far a k1-sparse vector θ has to be from the set of k0-sparse vectors so that a test is able to reject the null hypothesis with high probability. Two scenarios are considered. In the independent scenario, the covariates are i.i.d. normally distributed and the noise level is known. In the general scenario, both the covariance matrix of the covariates and the noise level are unknown. Although the minimax separation distances differ in these two scenarios, both of them actually depend on k0 and k1 illustrating that for this composite-composite testing problem both the size of the null and of the alternative hypotheses play a key role.


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Alexandra Carpentier. Nicolas Verzelen. "Optimal sparsity testing in linear regression model." Bernoulli 27 (2) 727 - 750, May 2021.


Received: 1 January 2019; Revised: 1 October 2019; Published: May 2021
First available in Project Euclid: 24 March 2021

Digital Object Identifier: 10.3150/20-BEJ1224

Keywords: high dimensional regression , minimax composite-composite testing , model testing

Rights: Copyright © 2021 ISI/BS


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Vol.27 • No. 2 • May 2021
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