Extended Hotelling $$T^2$$ test in distributed frameworks

Hypothesis test for a mean vector is a classical problem in data analysis but has been highly underinvestigated in distributed frameworks where samples of size n are located on k local sites. This paper focuses on the one-sample mean test, proposing synthesized test statistics with a much lower communication cost than the centralized Hotelling \(T^2\) test. For the homogeneous case, where data on different local sites are independent and identically distributed, the efficiency of our proposed test is comparable to that of the centralized one, and much better than the test constructed from the divide and conquer method. Besides, three heterogeneous cases are considered, where the distributions of the data on local sites can be different. Heterogeneous cases are much more challenging because the local sample means and covariance matrices may be inconsistent estimators. We construct communication-efficient testing procedures for heterogeneous cases, and the power of the proposed test statistics is comparable to that of the centralized one under some conditions. Simulation results verify the effectiveness of the proposed testing procedures.