Abstract: For a set of dependent random variables, and without using stationary or strong mixing assumptions, we derive the asymptotic independence between their sums and maxima. Then, we apply this result to high-dimensional testing problems. Here, we combine the sum-type and max-type tests, and propose a novel test procedure for the one-sample mean test, two-sample mean test and regression coefficient test in a high-dimensional setting. Based on the asymptotic independence between the sums and maxima, we establish the asymptotic distributions of the test statistics. Simulation studies show that our proposed tests perform well regardless of the sparsity of the data. Examples based on real data are also presented to demonstrate the advantages of our proposed methods.

Key words and phrases: Asymptotic normality, asymptotic independence, extreme-value distribution, high-dimensional tests, large p and small n.