Superconsistency of Tests in High Dimensions

TL;DR

This paper investigates the limitations of aggregate tests in high-dimensional data, demonstrating that no test can significantly outperform the likelihood ratio test in terms of consistency for most alternatives.

Contribution

It provides an impossibility result showing that superconsistency points for any test are asymptotically negligible in high dimensions.

Findings

Likelihood ratio test is asymptotically optimal.

Superconsistency points are negligible in high-dimensional settings.

No test can substantially outperform the likelihood ratio test in the limit.

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.