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Superconsistency of Tests in High Dimensions

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  • Anders Bredahl Kock
  • David Preinerstorfer

Abstract

To assess whether there is some signal in a big database, aggregate tests for the global null hypothesis of no effect are routinely applied in practice before more specialized analysis is carried out. Although a plethora of aggregate tests is available, each test has its strengths but also its blind spots. In a Gaussian sequence model, we study whether it is possible to obtain a test with substantially better consistency properties than the likelihood ratio (i.e., Euclidean norm based) test. We establish an impossibility result, showing that in the high-dimensional framework we consider, the set of alternatives for which a test may improve upon the likelihood ratio test -- that is, its superconsistency points -- is always asymptotically negligible in a relative volume sense.

Suggested Citation

  • Anders Bredahl Kock & David Preinerstorfer, 2021. "Superconsistency of Tests in High Dimensions," Papers 2106.03700, arXiv.org, revised Jan 2022.
    • Handle: RePEc:arx:papers:2106.03700
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    References listed on IDEAS

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    1. Ruth Heller & Nilanjan Chatterjee & Abba Krieger & Jianxin Shi, 2018. "Post-Selection Inference Following Aggregate Level Hypothesis Testing in Large-Scale Genomic Data," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(524), pages 1770-1783, October.
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    3. Anders Bredahl Kock & David Preinerstorfer, 2019. "Power in High‐Dimensional Testing Problems," Econometrica, Econometric Society, vol. 87(3), pages 1055-1069, May.
    4. Joseph P. Romano & Michael Wolf, 2005. "Exact and Approximate Stepdown Methods for Multiple Hypothesis Testing," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 94-108, March.
    5. Paul R. Rosenbaum, 2008. "Testing hypotheses in order," Biometrika, Biometrika Trust, vol. 95(1), pages 248-252.
    6. Ruth Heller & Amit Meir & Nilanjan Chatterjee, 2019. "Post‐selection estimation and testing following aggregate association tests," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(3), pages 547-573, July.
    7. Nicolai Meinshausen, 2008. "Hierarchical testing of variable importance," Biometrika, Biometrika Trust, vol. 95(2), pages 265-278.
    8. Yekutieli, Daniel, 2008. "Hierarchical False Discovery RateControlling Methodology," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 309-316, March.
    9. Jianqing Fan & Yuan Liao & Jiawei Yao, 2015. "Power Enhancement in High‐Dimensional Cross‐Sectional Tests," Econometrica, Econometric Society, vol. 83(4), pages 1497-1541, July.
    10. Vladimir Vovk & Ruodu Wang, 0. "Combining p-values via averaging," Biometrika, Biometrika Trust, vol. 107(4), pages 791-808.
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    Cited by:

    1. Thilo Reinschlussel & Martin C. Arnold, 2024. "Information-Enriched Selection of Stationary and Non-Stationary Autoregressions using the Adaptive Lasso," Papers 2402.16580, arXiv.org.

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