Abstract: For high-dimensional data with a small sample size, we cannot use Hotelling’s T² test to test the mean vectors because of the singularity problem in the sample covariance matrix. To overcome this problem, there are three main approaches but each has limitations and only works well in certain situations. Inspired by this, we propose a pairwise Hotelling method for testing high-dimensional mean vectors that provides a good balance between the existing approaches. To use the correlation information efficiently, we construct the new test statistics as the sum of Hotelling’s test statistics for the covariate pairs with strong correlations and the squared t-statistics for the individual covariates that have little correlation with others. We further derive the asymptotic null distributions and power functions for the proposed tests under some regularity conditions. Numerical results show that our tests are able to control the type-I error rates and achieve a higher statistical power than that of existing methods, especially when the covariates are highly correlated. Two real-data examples are used to demonstrate the efficacy of our pairwise Hotelling’s tests.

Key words and phrases: High-dimensional data, Hotelling’s test, pairwise correlation, screening, statistical power, type-I error rate.