Tests for the mean of high-dimensional data

TL;DR

This paper introduces a new high-dimensional mean test based on a statistic that avoids covariance matrix inversion, with proven asymptotic properties and bootstrap validity, applicable without sparsity assumptions.

Contribution

It proposes a novel test statistic for high-dimensional means that is computationally feasible and theoretically justified, extending classical results to infinite-dimensional settings.

Findings

The test statistic V_n = n||X_n||^2 has a well-defined asymptotic distribution.

Bootstrap approximation is valid for the distribution of the test statistic.

The method does not require sparsity or structural assumptions on the covariance matrix.

Abstract

We consider the problem of testing the mean of high-dimensional data when the dimension may grow without explicit rate restrictions relative to the sample size. The proposed procedure is based on the statistic V_n = n||Xn||^2, which avoids inversion of the covariance matrix and is therefore suitable for high-dimensional settings.We establish asymptotic distributional results for both fixed and increasing dimension by embedding the observations into the Hilbert space l2. Furthermore, we prove the asymptotic validity of a bootstrap approximation for the distribution of the test statistic. The resulting bootstrap test yields asymptotic level-a procedures without requiring sparsity assumptions or structural conditions on the covariance matrix. In all this, a new Central Limit Theorem in l2 is proving to be an extremely useful tool.