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Superconsistency of Tests in High Dimensions

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.

Superconsistency of Tests in High Dimensions

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.

Paper Structure

This paper contains 10 sections, 4 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Framework and terminology
  3. Superconsistency points
  4. Improving on the LR test
  5. The relative volume of the set of superconsistency points
  6. $p$-norm based tests
  7. Unrestricted sequences of tests
  8. Conclusion
  9. Proof of Theorem \ref{['thm:nogain2']}
  10. Proof of Proposition \ref{['thm:nogain3']}

Key Result

Theorem \oldthetheorem

Let $\kappa_{d,2}$ be a sequence of critical values such that the asymptotic size of $\{2, \kappa_{d,2}\}$ is $\alpha \in (0, 1)$. Then

Figures (1)

  • Figure 1: Illustration of $\mathbb{B}_2^d(r_d)$ (red), $\mathbb{D}_d \subseteq \mathbb{B}_2^d(r_d)$ and $\mathbb{D}_d$ (green) for $d = 2$.

Theorems & Definitions (10)

  • Remark 2.1
  • Theorem \oldthetheorem
  • Remark 3.1
  • Remark 3.2
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Remark 5.1: Spherical measure instead of relative volume
  • Proposition \oldthetheorem
  • Remark 5.2