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April 2022 Testability of high-dimensional linear models with nonsparse structures
Jelena Bradic, Jianqing Fan, Yinchu Zhu
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Ann. Statist. 50(2): 615-639 (April 2022). DOI: 10.1214/19-AOS1932


Understanding statistical inference under possibly nonsparse high-dimensional models has gained much interest recently. For a given component of the regression coefficient, we show that the difficulty of the problem depends on the sparsity of the corresponding row of the precision matrix of the covariates, not the sparsity of the regression coefficients. We develop new concepts of uniform and essentially uniform nontestability that allow the study of limitations of tests across a broad set of alternatives. Uniform nontestability identifies a collection of alternatives such that the power of any test, against any alternative in the group, is asymptotically at most equal to the nominal size. Implications of the new constructions include new minimax testability results that, in sharp contrast to the current results, do not depend on the sparsity of the regression parameters. We identify new tradeoffs between testability and feature correlation. In particular, we show that, in models with weak feature correlations, minimax lower bound can be attained by a test whose power has the n rate, regardless of the size of the model sparsity.

Funding Statement

The first author was supported by NSF Grant DMS-1712481. The second author was supported by NSF Grants DMS-1662139 and DMS-DMS-1712591 and NIH Grants 5R01-GM072611-12.


Download Citation

Jelena Bradic. Jianqing Fan. Yinchu Zhu. "Testability of high-dimensional linear models with nonsparse structures." Ann. Statist. 50 (2) 615 - 639, April 2022.


Received: 1 November 2018; Revised: 1 November 2019; Published: April 2022
First available in Project Euclid: 7 April 2022

Digital Object Identifier: 10.1214/19-AOS1932

Primary: 62C20 , 62F03
Secondary: 62F30 , 62J07

Keywords: confidence intervals , ℓ2-constraint , minimax theory , uniform nontestability

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 2 • April 2022
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