Abstract: For the same null hypothesis, there usually exist multiple valid test statistics. In nearly all cases, any individual statistic is only powerful against specific types of alternatives, and could be rather weak in picking up signals of other types. It is thus crucial, especially in high-dimensional settings, to combine the information contained in different test statistics in order to maintain robust power against a wide range of alternatives, thus avoiding the worst-case scenario. Methods have been proposed for similar purposes, but they are either computationally expensive or lack theoretical justification. In this paper, we present a general and easy-to-implement procedure for fusing multiple valid statistics using resampling methods, such as bootstrap or permutation. The consistency of this procedure is proved for three popular high-dimensional hypothesis testing problems. The results of numerical studies show that this fusion procedure maintains robust performance against a wide range of alternatives, whereas individual test statistics often suffer from extremely low power.

Key words and phrases: Consistency of test, high-dimensional data, independence test, permutation, two-sample mean comparison.