Testing the Mean Vector for High-Dimensional Data

In one-sample mean testing for high-dimensional data, existing tests, e.g., Chen and Qin (Ann Stat 38(2):808–835, 2010) and Wang et al. (J Am Stat Assoc 110(512):1658–1669, 2015), assume that the data are either normally distributed or from a latent factor model. In this paper, we remove these restrictive assumptions and develop a new asymptotic theory, showing that the asymptotic null distribution is a mixture of χ2 mixture and normal distributions. With more conditions on the eigenvalues of the covariance matrices, a normal or χ2 mixture approximation for the limiting null distribution is derived. The power functions of two test statistic under high-dimensional version local and fixed alternative are also analyzed. A wild bootstrap procedure is proposed to determine the critical values of the mixture of χ2 mixture and normal distributions, which is easy to implement and fast to run. Numerical simulations show that our proposed methods control the test’s size more precisely than existing methods using the normal approximation. The merit of the proposed methods is further demonstrated on a real data example.