Projection-based High-dimensional Sign Test

This article is concerned with the high-dimensional location testing problem. For high-dimensional settings, traditional multivariate-sign-based tests perform poorly or become infeasible since their Type I error rates are far away from nominal levels. Several modifications have been proposed to address this challenging issue and shown to perform well. However, most of modified sign-based tests abandon all the correlation information, and this results in power loss in certain cases. We propose a projection weighted sign test to utilize the correlation information. Under mild conditions, we derive the optimal direction and weights with which the proposed projection test possesses asymptotically and locally best power under alternatives. Benefiting from using the sample-splitting idea for estimating the optimal direction, the proposed test is able to retain type-I error rates pretty well with asymptotic distributions, while it can be also highly competitive in terms of robustness. Its advantage relative to existing methods is demonstrated in numerical simulations and a real data example.