Testability of high-dimensional linear models with non-sparse structures

Jelena Bradic Jianqing Fan Yinchu Zhu

Statistics Theory and Methods mathscidoc:1912.43418

arXiv preprint arXiv:1802.09117, 2018.2
There is gained interest in understanding statistical inference under possibly non-sparse high-dimensional models. 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 non-testability that allow the study of limitations of tests across a broad set of alternatives. Uniform non-testability 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 of the test. Implications of the new constructions include new minimax testability results that in sharp contrast to the existing 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 parametric rate regardless of the size of the model sparsity.
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@inproceedings{jelena2018testability,
  title={Testability of high-dimensional linear models with non-sparse structures},
  author={Jelena Bradic, Jianqing Fan, and Yinchu Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114316752142978},
  booktitle={arXiv preprint arXiv:1802.09117},
  year={2018},
}
Jelena Bradic, Jianqing Fan, and Yinchu Zhu. Testability of high-dimensional linear models with non-sparse structures. 2018. In arXiv preprint arXiv:1802.09117. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114316752142978.
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