Tests for the mean of high-dimensional data

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.