High-dimensional general linear hypothesis testing under heteroscedasticity

- A general linear hypothesis testing problem on mean vectors of several populations is considered.
- A U-statistic based test statistic is proposed but it can be quickly calculated.
- Asymptotic normality and power of the test are derived under mild conditions.
- The test is applicable to non-normal heteroscedastic multi-sample problems.
- An efficient and consistent estimator of the test statistic's variance is provided.

In recent years, high-dimensional data has become increasingly prevalent with rapid development of data collecting technologies. Much work has been done for hypothesis testing on mean vectors, especially for high-dimensional two-sample problems. Rather than considering a specific problem, we are interested in a general linear hypothesis testing problem on mean vectors of several populations, which includes many existing hypotheses about mean vectors as its special cases. We propose a test statistic based on a linear combination of U-statistics which can be quickly calculated without using computationally intensive U-statistics. Asymptotic normality and power of our test are derived under mild conditions without requiring an explicit relation between the data dimension and the sample size. Our test is applicable to non-normal multi-sample high-dimensional data without assuming a common covariance matrix among different samples. It also works well even when different samples follow different distributions, provided that some moment conditions are satisfied. A simple, computationally-efficient, and ratio-consistent estimator of the unknown variance of our test statistic is also provided which is not based on U-statistic calculations. Simulation studies and a real data example are presented to demonstrate the good performance of our test.