Testing the Mean Vector for High-Dimensional Data

Abstract

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 $\chi ^2$ mixture and normal distributions. With more conditions on the eigenvalues of the covariance matrices, a normal or $\chi ^2$ 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 $\chi ^2$ 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.

Funding

National Natural Science Foundation of China(12271370)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature