An Inverse Norm Sign Test of Location Parameter for High-Dimensional Data

We consider the one sample location testing problem for high-dimensional data, where the data dimension is potentially much larger than the sample size. We devise a novel inverse norm sign test (INST) that is consistent and has much improved power than many existing popular tests. We further construct a general class of weighted spatial sign tests which includes these existing tests, and show that INST is the optimal member within this class, in that it is consistent and is uniformly more powerful than all other members. We establish the asymptotic null distribution and local power property of the class of tests rigorously. Extensive numerical experiments demonstrate the superiority of INST in terms of both efficiency and robustness.